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Punjab State Board PSEB 11th Class Biology Book Solutions Chapter 6 Anatomy of Flowering Plants Textbook Exercise Questions and Answers.

PSEB Solutions for Class 11 Biology Chapter 6 Anatomy of Flowering Plants

PSEB 11th Class Biology Guide Anatomy of Flowering Plants Textbook Questions and Answers

Question 1.
State the location and function of different types of meristems.
Answer:
A meristematic tissue represents a group of cells that have retained the power of division throughout the life of an individual. The meristematic tissues are of three types, i.e., apical, intercalary and lateral meristem.

• Apical Meristem: These meristems are present at the apices of shoot and roots of the plants. Apical meristems are responsible for the increase in length of all the primary tissues.
• Intercalary Meristem: It is the meristem that occurs between the mature tissues. They occur in grasses and regenerate parts removed by the grazing herbivores. It contributes to the formation of the primary plant body.
• Lateral Meristem: It occurs in the mature regions of roots arid shoots of many plants, particularly that produce woody axis. These meristems are responsible for producing the secondary tissues.

Question 2.
Cork cambium forms tissues that form the cork. Do you agree with this statement? Explain.
Answer:
Yes, cork cambium forms tissues that form cork. As the stem continues to increase in girth another meristematic tissue called cork cambium or phellogen develops in cortex region of stem. The phellogen cuts off cells on both sides. The outer cells differentiate into cork or phellem. The inner cells differentiate into secondary cortex or phelloderm. Cork is impervious to water due to suberin and provides protection to underlying tissues.

Question 3.
Explain the process of secondary growth in the stems of woody angiosperms with the help of schematic diagrams. What is its significance?
Answer:
Secondary growth in stems of woody angiosperms occur by two types of cambia, i.e., vascular cambium and cork cambium.
(i) Vascular Cambium: Certain cells of medullary rays become meristematic to form interfascicular cambium. The fascicular cambium and the interfascicular cambium join to form a complete ring called cambial ring. The cells of the cambial ring undergo mitotic divisions and produce secondary phloem on its outer side and secondary xylem on its inner side.

At places, vascular cambium possesses ray initials. They form vascular rays, phloem rays in secondary phloem and wood rays in secondary xylem.

As new secondary phloem becomes functional, the previous older phloem gets crushed. Secondary xylem or wood persists. As a result wood grows

with age in the form of annual rings. In each annual ring, there is wide or broader spring or early wood or spring wood and narrow autumn or late wood.

In old stems, the central part of wood becomes non-functional and dark coloured due to tyloses and deposit of resins, gums, tannins. It is called r duramen or heartwood. The outer, functional wood is called sapwood.

(ii) Cork Cambium: As the stem continues to increase in girth due to the activity of vascular cambium the outer cortical and epidermal layers get broken and need to be replaced to provide new protective cell layers. In this way, cork cambium or phellogen develops in the cortex region. Phellogen cuts of cells on both sides.

The outer cells differentiate into cork or phellem while, the inner cells ; differentiate into secondary cortex or phelloderm. Due to the activity of cork cambium, pressure builds up on remaining layers peripheral to
phellogen and ultimately these layers die and slough off. At places, aerating pores called lenticels develop, which have loosely arranged , complementary cells.

Significance of Secondary Growth

• It replaces old non-functional tissues.
• It provides fire proof, insect proof and insulating cover around the older plant parts.
• Commercial cork is a product of secondary growth.
• Wood is the product of secondary growth.

Question 4.
Draw illustrations to bring out the anatomical difference between:
(a) Monocot root and Dicot root
(b) Monocot stem and Dicot stem
Answer:

(a) Anatomical Differences between Monocot Root and Dicot Root
(i) Anatomy of Monocot Root
(a) The structure of epidermis, cortex, endodermis and pericycle of a monocot root resembles exactly those of a dicot root.
(b) Vascular bundles are radial, and polyarch.
(c) Pith is large and well-developed; it is composed of parenchyma cells

(ii) Anatomy of dicot root

T.S of Dicot root
(a) Epidermis is single layered and many cells bear root hairs; cuticle is absent.
(b) Cortex is made of several layers of parenchyma cells.
(c) Endodermis consists of a single-layer of cells. The cells have a deposition of suberin, in the form of casparian strips, on their radial and tangential walls.
(d) Pericycle comprises a few layers of specialised parenchyma cells inner to the endodermis.
(e) Vascular bundles are radial and may range between two and six, though commonly there are four groups; i.e., tetrarch; xylem is exarch.
(f) Pith is very small and made of parenchyma cells.

(b) Anatomical Differences between Monocot Stem and Dicot stem
(i) Anatomy of monocot stem
(a) Epidermis is single layered and trichomes are absent; cuticle is present on its outer surface.
(b) Hypodermis consists of two or three layers of sclerenchyma cells.
(c) Ground tissue is parenchymatous and is not differentiated into cortex or pith.
(d) Vascular bundles are many and scattered in the ground tissue; they vary in size.

T.S. of Monocot Stem

• Each vascular bundle is surrounded by sclerenchymatous bundle sheath.
• The vascular bundles are conjoint and closed; xylem is endarch and characteristically a protoxylem lacuna is present.

(ii) Anatomy of dicot stem
(a) Epidermis is the outermost layer of cells; externally it is covered with a cuticle and may bear trichomes and a few stomata.
(b) Hypodermis consists of a few layers of collenchyma cells, just below the epidermis.
(c) Cortex consists of parenchyma cells.
(d) Endodermis is single layered and the cells are rich in starch grains and hence it is also referred to as starch sheath.
(e) Pericyde occurs inner to the endodermis, above the phloem of vascular bundles in the form of semi-lunar patches (hence also referred to as bundle caps); it is composed of sclerenchyma.
T.S. of Dicot Stem

(f) Vascular bundles are characteristically arranged in the form of a ring.

• Each vascular bundle is conjoint and open with intra-fascicular cambium; xylem is endarch.
  (g) Medullary rays are the few layers of radially placed parenchyma cells, in between the vascular bundles.
  (h) Pith is composed of parenchyma cells.

Question 5.
Cut a transverse section of young stem of a plant from your school garden and observe it under the microscope. How would you ascertain whether it is a monocot stem or a dicot stem? Give reasons.
Answer:
Transverse section of a monocot stem possess following characters:

• Dumbbell-shaped guard cells in stomata in epidermis.
• Sclerenchymatous hypodermis.
• No concentric arrangement of internal tissues.
• Unifrom ground tissue showing no tissue differentiation.
• More than 8 scattered vascular bundles.
• Bundle sheath is present.
• No secondary growth normally.
• Xylem vessels arranged in Y-shaped manner.
• Protoxylem cavity usually present in vascular tissues.

Transverse section of a dicot stem possess following characters:

• Kidney-shaped guard cells in stomata present in epidermis.
• Collenchymatous hypodermis.
• Concentric arrangement of internal tissues.
• Differentiation of ground tissue into cortex, endodermis, pericycle and pith.
• The vascular bundles are arranged in a ring.
• Conjoint, collateral and open vascular bundles.
• Without bundle sheath.
• Secondary growth takes place.
• Xylem vessels arranged in rows.

[Note: For figures refer to Q.No. 4]

Question 6.
The transverse section of a plant material shows the following anatomical features:
(a) The vascular bundles are conjoint, scattered and surrounded by a sclerenchymatous bundle sheaths.
(b) Phloem parenchyma is absent. What will you identify it as?
Answer:
It is a transverse section of monocotyledonous stem.

Question 7.
Why are xylem and phloem called complex tissues?
Answer:
Xylem and phloem are made up of more than one type of cells that is why they are called as complex tissues :
(i) Xylem is composed of four different kinds of elements, namely-tracheids, vessels, xylem fibres and xylem parenchyma. Tracheids are dead tube-like cells which are thick walled, vessels are made up of large number of tube cells placed end to end. Xylem fibres are thick walled cells that maybe septate and aseptate. Xylem parenchyma is living and thin walled cells.

(ii) Phloem is composed of sieve tube elements, companion cells, phloem parenchyma and phloem fibres. Sieve tube elements are tube-like cells, whereas phloem parenchyma are living cells and phloem fibres are thick walled lignified cells.

Question 8.
What is stomatal apparatus? Explain the structure of stomata with a labelled diagram.
Answer:
The minute pores present in the epidermis are known as stomata. The stomata may be surrounded by either bean-shaped (in dicots) or by dumb-bell-shaped (in monocots) guard cells. The guard cells in turn are surrounded by other epidermal cells, which are known as subsidiary or

accessory cells. The stomatal aperture, guard cells, accessory cells constitute the stomatal apparatus.

Question 9.
Name the three basic tissue systems in the flowering plants. Give the tissue names under each system.
Answer:
The three basic; tissue systems in flowering plants are as follows:

• Epidermal Tissue System: The tissue related to this system are epidermis, cuticle and wax, stomata and trichomes.
• Ground Tissue System: It consists of cortex, endodermis, pericycle, medullary rays, pith and ground tissue of leaves.
• Vascular Tissue System: It contains conducting tissues like xylem and phloem.

Question 10.
How is the study of plant anatomy useful to us?
Answer:
Plant anatomy is the study of internal structure of living organisms.

• It describes the tissues involved in assimilation of food and its storage, transportation of water, i.e., xylem tissue, transportation of minerals. i.e., phloem and those involved in providing mechanical support to the plant,
• Study of internal structure of plants helps to understand their adaptations of diverse environments.
• The study of plant anatomy also help in understanding the functional organisation of higher plants.

Question 11.
What is periderm? How does periderm formation take place in
the dicot stems?
Answer:
Phellogen, phellem and phelloderm are collectively called as periderm. Phellogen develops usually in the cortex region. Phellogen is a couple of layers thick. Phellogen cuts off cells on both sides. The outer cells 1 differentiate into cork or phellem, while the inner cells differentiate into secondary cortex or phelloderm. All these together form periderm.

Question 12.
Describe the internal structure of a dorsiventral leaf with the help of labelled diagrams.
Answer:
The vertical section of a dorsiventral leaf through the lamina shows three ; main parts namely, epidermis, mesophyll and vascular system. The epidermis which covers both the upper surface (adaxial epidermis) and lower surface (abaxial epidermis) of the leaf has a conspicuous cuticle. The abaxial epidermis generally bears more stomata than the adaxial epidermis. The latter may even lack stomata. The tissue between the upper and the lower epidermis is called the mesophyll. Mesophyll, which possesses chloroplasts and carry out photosynthesis, is made up of parenchyma. It has two types of cells – the palisade parenchyma and the spongy parenchyma.

The adaxially placed palisade parenchyma is made up of elongated cells, which are arranged vertically and parallel to each other. The oval or round and loosely arranged spongy parenchyma is situated below the palisade cells and extends to the lower epidermis. There are numerous large spaces and air cavities between these cells. Vascular system includes vascular bundles, which can be seen in the veins and the midrib. The size of the vascular bundles are dependent on the size of the veins. The veins vary in thickness in the reticulate venation of the dicot leaves. The vascular bundles are surrounded by a layer of thick walled bundle sheath cells.
T.S. of Dorsiventral Leaf

Punjab State Board PSEB 11th Class Biology Book Solutions Chapter 17 Breathing and Exchange of Gases Textbook Exercise Questions and Answers.

PSEB Solutions for Class 11 Biology Chapter 17 Breathing and Exchange of Gases

PSEB 11th Class Biology Guide Breathing and Exchange of Gases Textbook Questions and Answers

Question 1.
Define vital capacity. What is its significance?
Answer:
Vital Capacity (VC): The maximum volume of air a person can breathe in after a forced expiration is called vital capacity. Vital capacity is higher in athletes and singers. Vital capacity shows the strength of our inspiration and expiration.

Question 2.
State the volume of air remaining in the lungs after a normal breathing.
Answer:
The volume of air remaining in the lungs even after a forcible expiration averages 1100 ml to 1200 ml.

Question 3.
Diffusion of gases occurs in the alveolar region only and not in the other parts of respiratory system. Why?
Answer:
Alveoli are the primaty sites of gas exchange in the respiratory system. Exchange of gases occur between blood and these tissues. O₂ and CO₂ are exchanged in these sites by simple diffusion mainly based on pressure/concentration gradient. The diffusion membrane for gas exchange is made up of three major layers.
These layers are :

1. Squamous epithelium of alveoli.
2. Endothelium of alveolar capillaries.
3. Basement substance in between them. Its total thickness is much less than a millimetre. Therefore, all the factors in our body are favourable for diffusion of O₂ from alveoli to tissues and that of CO₂ from tissues to alveoli.

Question 4.
What are the major transport mechanisms for CO₂? Explain.
Answer:
Transport of Carbon Dioxide: CO₂ in gaseous form diffuses out of the cells into capillaries, where it is transported in following ways :
(i) Transport in dissolved form: About 7% CO₂ is carried in dissolved form through the plasma because of its high solubility.
(ii) Transport as bicarbonate: The largest fraction (about 70%) is carried in plasma as bicarbonate ions (HCO₃). At the tissues site, where pCO₂ is high due to catabolism, CO₂ diffuses into the blood (RBCs and plasma) and forms HCO₃^– and H^–.

This reaction is faster in RBCs because they contain an enzyme carbonic anhydrase. Hydrogen ion released during the reaction bind to Hb, triggering the Bohr effect.
At the alveolar site, where pCO₂ is low, the reaction proceeds in opposite direction forming CO₂ and H₂O. Thus, CO₂ trapped as bicarbonate at tissue level and transported to alveoli is released as CO₂.

(iii) Transport as carbaminohaemoglobin: Nearly 20-25% CO₂ is carried by haemoglobin as carbaminohaemoglobin, CO₂ entering the blood combines with the NH₂ group of the reduced Hb.

The reaction releases oxygen from oxyhaemoglobin.
Factors affecting the binding of CO₂ and Hb are as follows:

• Partial pressure of CO₂.
• Partial pressure of O₂ (major factor).

In tissues, pCO₂ is high and pO₂ is low, more binding of CO₂ occurs while, in the alveoli, pCO₂ is low and pO₂ is high, dissociation of CO₂ from HbCO₂ takes place, i.e., CO₂, which is bound to Hb from the tissues is delivered at the alveoli.

Question 5.
What will be the pO₂ and pCO₂, in the atmospheric air compared to those in the alveolar air?
(i) pO₂ lesser and pCO₂ higher
(ii) pO₂ higher and pCO₂ lesser
(iii) pO₂ higher and pCO₂ higher
(iv) pO₂ lesser and pCO₂ lesser
Answer:
(i) In the alveolar tissues, where low pO₂, high pCO₂, high H⁺ concentration, these conditions are favourable for dissociation of oxygen from the oxyhaemoglobin.

(ii) When there is high pO₂, low pCO₂, less H⁺ concentration and lesser temperature, the factors are all favourable for formation of oxyhaemoglobin.

(iii) When pO₂ is high in the alveoli and pCO₂ is high in the tissues then the oxygen diffuses into the blood and combines with oxygen forming oxyhaemoglobin and CO₂ diffuses out.

(iv) When pO₂ is low in the alveoli and pCO₂ is low in the tissues then these conditions are favourable for dissociation of oxygen from the oxyhaemoglobin.

Question 6.
Explain the process of inspiration under normal conditions.
Answer:
Inspiration is the process during which atmospheric air is drawn in. Inspiration is initiated by the contraction of diaphragm, which increases the volume of thoracic chamber in the anteroposterior axis. The contraction of external intercostal muscles lifts up the ribs and the sternum causing an increase in the volume of the thoracic chamber in the dorso-ventral axis.

The overall increase in the thoracic volume causes a similar increase in pulmonary volume. An increase in pulmonary volume decreases the intrapulmonary pressure to less than the atmospheric pressure, which forces the air from outside to move into the lungs, i. e., inspiration. On an average, a healthy human breathes 12-16 times/minute.

Question 7.
How is respiration regulated?
Answer:
Respiratory rhythm centre is primarily responsible for regulation of respiration. This centre is present in the medulla.
Pneumotaxic centre, present in the pons region, also coordinates respiration. Apart from them, receptors associated with aortic arch and carotid artery, can also recognize changes in CO₂ and H⁺ concentration and send signal to the rhythm centre for proper action.

Question 8.
What is the effect of pCO₂ on oxygen transport?
Answer:
Partial pressure of CO₂ (pCO₂) can interfere the binding of oxygen with haemoglobin, i.e., to form oxyhaemoglobin.
(i) In the alveoli, where there is high pO₂ and low pCO₂, less H⁺ concentration and low temperature, more formation of oxyhaemoglobin occur.

(ii) In the tissues, where low pO₂, high pCO₂, high H⁺ concentration and high temperature exist, the conditions are responsible for dissociation of oxygen from the oxyhaemoglobin.

Question 9.
What happens to the respiratory process in a man going up a hill?
Answer:
At hills, the pressure of air falls and the person cannot get enough oxygen in the lungs for diffusion in blood. Due to deficiency of oxygen, the person feels breathlessness, headache, dizziness, nausea, mental fatigue and a bluish colour on the skin, nails and lips.

Question 10.
What is the site of gaseous exchange in an insect?
Answer:
The actual site of gaseous exchange in an insect is tracheoles and tracheolar end cells.

Question 11.
Define oxygen dissociation curve. Can you suggest any reason for its sigmoidal pattern?
Answer:
A sigmoid curve is obtained when percentage saturation of haemoglobin with O₂ is plotted against the pO₂.
This curve is called the oxygen dissociation curve and is highly useful in studying the effect of factors like pCO₂, H⁺ concentration, etc., on binding of O₂ with haemoglobin.

In the alveoli, where there is high pO₂, low pCO₂, lesser H⁺ concentration and lower temperature, the factors are all favourable for the formation of oxyhaemoglobin, whereas in the tissues, where low pO₂, high pCO₂, high H⁺ concentration and higher temperature exist, the conditions are favourable for dissociation of oxygen from the oxyhaemoglobin. This dearly indicates that O₂ gets bound to haemoglobin in the lung surface and gets dissociated at the tissues. Every 100 ml of oxygenated blood can deliver around 5 ml of O₂ to the tissues under normal physiological conditions.

Question 12.
Have you heard about hypoxia? Try to gather information about it, and discuss with your friends.
Answer:
Hypoxia is the shortage of oxygen supply to the blood due to :
(a) normal shortage in air
(b) oxygen deficiency on high mountains (mountain sickness), anaemia and phytotoxicity or poisoning of electron transport system.

Question 13.
Distinguish between:
(a) IRV and ERV
(b) Inspiratory Capacity and Expiratory Capacity
(c) Vital Capacity and Total Lung Capacity
Answer:
(a) IRV and ERV Inspiratory Reserve Volume (IRV): Additional volume of air, a person can inspire by a forcible inspiration. This is about 2500-3000 mL. Expiratory Reserve Volume (ERV): Additional volume of air, a person can expire by a forcible expiration. This is about 1000-1100 mL.

(b) Inspiratory Capacity and Expiratory Capacity Inspiratory Capacity (IC): Total volume of air a person can inspire after a normal expiration. This includes tidal volume and inspiratory reserve volume (TV+IRV).
Expiratory Capacity (EC): Total volume of air a person can expire after a normal inspiration. This includes tidal volume and expiratory reserve volume (TV+ERV)

(c) Vital Capacity and Total Lung Capacity
Vital Capacity (VC): The maximum volume of air, a person can breathe in after a forced expiration. This includes ERV, TV and IRV or the maximum volume of air a person can breathe out after a forced inspiration.
Total Lung Capacity (TLC): Total volume of air accommodated in the lungs at the end of a forced inspiration. This includes RV, ERV, TV and IRV or vital capacity + residual volume.

Question 14.
What is tidal volume? Find out the tidal volume (approximate value) for a healthy human in an hour.
Answer:
Tidal Volume (TV): Volume of air inspired or expired during a normal respiration is called tidal volume. It is about 500 mL., i.e., a healthy man can inspire or expire approximately 6000 to 8000 mL of air per minute.

Punjab State Board PSEB 11th Class Biology Book Solutions Chapter 18 Body Fluids and Circulation Textbook Exercise Questions and Answers.

PSEB Solutions for Class 11 Biology Chapter 18 Body Fluids and Circulation

PSEB 11th Class Biology Guide Body Fluids and Circulation Textbook Questions and Answers

Question 1.
Name the components of the formed elements in the blood and mention one major function of each of them.
Answer:
(a) Erythrocytes: They are also known as Red Blood Cells (RBC). They are the most abundant of all the cells in blood. A healthy’adult man has, on an average, 5 millions to 5.5 millions of RBCs mm ^-3 of blood. RBCs are formed in the red bone marrow in the adults.
RBCs are devoid of nucleus in most of the mammals and are biconcave in shape. They have a red coloured, iron containing complex protein called haemoglobin. A healthy individual has 12-16 gms of haemoglobin in every 100 ml of blood.
these molecules play a significant role in transport of respiratory gases. RBCs have an average life span of 120 days after which they are destroyed in the spleen. Hence, spleen is also known as the graveyard of RBCs.

(b) Leucocytes: They are also known as White Blood Cells (WBC) as they are colourless due to the lack of haemoglobin. They are nucleated and are relatively lesser in number which averages 6000-8000 mm^-3 of blood. Leucocytes are generally short-lived.

There are two main categories of WBCs :
1. Granulocytes, e.g., neutrophils, eosinophils and basophils
2. Agranulocytes. e.g., lymphocytes and monocytes.
Neutrophils are the most abundant cells (60-65 per cent) of the total WBCs and basophils are the least (0.5-1 per cent) among them. Neutrophils and monocytes (6-8 per cent) are phagocytic cells which destroy foreign organisms entering the body.

Basophils secrete histamine, serotonin, heparin, etc., and are involved in inflammatory reactions. Eosinophils (2-3 per cent) resist infections and are also associated with allergic reactions. Lymphocytes (20-25 per cent) are of two major types- ‘B’ and T forms. Both B and T lymphocytes are responsible for immune responses of the body.

(c) Platelets: Platelets or thrombocytes, are involved in the coagulation or clotting of blood. A reduction in their number can lead to clotting disorders, which will lead to excessive loss of blood from the body.

Question 2.
What is the importance of plasma proteins?
Answer:
Fibrinogen, globulins and albumins are the major plasma proteins. Fibrinogens are needed for clotting or coagulation of blood. Globulins primarily are involved in defense mechanisms of the body and the albumins help in osmotic balance.

Question 3.
Match column I with column II.

Column I	Column II
A. Eosinophils	1. Coagulation
B. RBC	2. Universal recipient
C. AB group	3. Resist infections
D. Platelets	4. Contraction of heart
E. Systole	5. Gas transport

Answer:

Column I	Column II
A. Eosinophils	3. Resist infections
B. RBC	5. Gas transport
C. AB group	2. Universal recipient
D. Platelets	1. Coagulation
E. Systole	4. Contraction of heart

Question 4.
Why do we consider blood as a connective tissue?
Answer:
Blood is a mobile connective tissue derived from mesoderm which consists of fibre-free fluid matrix, plasma and other cells. It regularly circulates in the body, takes part in the transport of materials.

Question 5.
What is the difference between blood and lymph?
Answer:
Differences between Blood and Lymph

Blood	Lymph
It is red in colour due to the presence of haemoglobin in red cells.	It is colourless as red blood cells are absent.
It consists of plasma, RBC, WBC and platelets.	It consists of plasma and less number of WBC.
Glucose concentration is low.	Glucose concentration is higher than blood.
Clotting of blood is a fast process.	Clotting of lymph is comparatively slow.
It transports materials from one organ to other.	It transports materials from tissue cells into the blood.
Flow of blood is fast.	Lymph flows very slowly.
Its plasma has more proteins, calcium and phosphorus.	Its plasma has less protein, calcium and phosphorus.
It moves away from the heart and towards the heart.	It moves in one direction, i. e., from tissues to sub-clavians.

Question 6.
What is meant by double circulation? What is its significance?
Answer:
Double Circulation: In double circulation, the blood passes twice through the heart during one complete cycle. Double circulation is carried out by two ways :
1. Pulmonary circulation,
2. Systemic circulation

Significance: In birds and mammals, oxygenated and deoxygenated blood received by the left and right atria respectively passes on to the ventricles of the same sides. The ventricles pump it out without mixing up, i.e., two separate circulatory pathways are present in these organisms. This is the importance of double circulation.

Question 7.
Write the differences between:
(a) Blood and lymph
(b) Open and closed system of circulation
(c) Systole and diastole
(d) P-wave and T-wave
Answer:
(a)

Blood	Lymph
It is red in colour due to the presence of haemoglobin in red cells.	It is colourless as red blood cells are absent.
It consists of plasma, RBC, WBC and platelets.	It consists of plasma and less number of WBC.
Glucose concentration is low.	Glucose concentration is higher than blood.
Clotting of blood is a fast process.	Clotting of lymph is comparatively slow.
It transports materials from one organ to other.	It transports materials from tissue cells into the blood.
Flow of blood is fast.	Lymph flows very slowly.
Its plasma has more proteins, calcium and phosphorus.	Its plasma has less protein, calcium and phosphorus.
It moves away from the heart and towards the heart.	It moves in one direction, i. e., from tissues to sub-clavians.

(b) Differences between Open and Closed Circulatory Systems

Open Circulatory System	Closed Circulatory System
1. It is present in arthropods and molluscs.	It is present in annelids and chordates.
2. Blood pumped by heart passes through large vessels into open spaces or body cavities called sinuses.	Blood pumped by the heart is circulated through a loosed network of blood vessels.
3. Flow of blood is not regulated precisely.	It is more advantageous as the blood flow is more precisely regulated.

(c) Differences between Systole and Diastole

Systole	Diastole
1. The contraction of the muscles of auricles and ventricles is called systole.	It is the relaxation of atria and ventricle muscle.
2. It increases the ventricular pressure causing the closure of tricuspid and bicuspid valves due to attempted backflow of blood into atria.	The ventricular pressure falls causing the closure of semilunar valves which prevent backflow of blood into the ventricle.
3. Systolic pressure is higher and occurs during ventricular contraction.	Diastolic pressure is lower and occurs during ventricular expansion.

(d) Differences between P-wave and T-wave

P-wave	T-wave
The P-wave represents the electrical excitation (or depolarisation) of the arrÍa, which leads to the contraction of both the arria.	The T-wave represents the return of the ventricles from excited to normal state (repolarisation). The end of the T-wave marks the end of systole.

Question 8.
Describe the evolutionary change in the pattern of heart among the vertebrates.
Answer:
The heart among the vertebrates shows different patterns of evolution. Different groups of animals have evolved different methods for this transport. All vertebrates possess a muscular chambered heart.
Fishes have a 2-chambered heart with an atrium and a ventricle.
Amphibians and the reptiles (except crocodiles) have a 3-chambered heart with two atria and a single ventricle.
In crocodiles, birds and mammals possess a 4-chambered heart with two atria and two ventricles.

In fishes, the heart pumps out deoxygenated blood which is oxygenated by the gills and supplied to the body parts from where deoxygenated blood is returned to the heart.

In amphibians and reptiles, the left atrium receives oxygenated blood from the gills/lungs/skin and the right atrium gets the deoxygenated blood from other body parts. However, they get mixed up in the single ventricle which pumps out mixed blood.

In birds and mammals, oxygenated and deoxygenated blood received by the left and right atria respectively passes on to the ventricles of the same sides. The ventricles pump it out without any mixing up, i. e., two separate circulatory pathways are present in these organisms, hence, these animals have double circulation.

Question 9.
Why do we call our heart myogenic?
Answer:
Heart is myogenic in origin because the cardiac impulse is initiated in our heart muscles.

Question 10.
The sino-atrial node is called the pacemaker of our heart. Why?
Answer:
The sino-atrial node of heart is responsible for initiating and maintaining the rhythmic activity, therefore it is known as pacemaker of the heart.

Question 11.
What is the significance of atrioventricular node and atrioventricular bundle in the functioning of heart?
Answer:
Atrioventricular Node (AVN): It is the mass of tissue present in the lower-left corner of the right atrium close to the atrioventricular septum. It is stimulated by the impulses that sweep over the atrial myocardium. It is too capable of initiating impulses that cause contraction but at slower rate than SA node.

Atrioventricular Bundle (AV Bundle): It is a bundle of nodal fibres, which continues from AVN and passes through the atria-ventricular septa to emerge on the top of interventricular septum. The AV bundle, bundle branches and Purkinje fibres convey impulses of contraction from the AV node to the apex of the myocardium. Here the wave of ventricular contraction begins, then sweeps upwards and outwards, pumping blood into the pulmonary artery and the aorta.
This nodal musculature has the ability to generate action potentials without any external stimuli.

Question 12.
Define a cardiac cycle and the cardiac output.
Answer:
Cardiac Cycle: The sequential event in the heart which is cyclically repeated is called the cardiac cycle. It consists of systole and diastole of both the atria and ventricles.

Cardiac Output: It is the volume of blood pumped out by each ventricle per minute and averages 5000 mL or 5 L in a healthy individual. The body has the ability to alter the stroke volume as well as the heart rate and thereby the cardiac output. For example, the cardiac output of an athlete will be much higher than that of an ordinary man.

Question 13.
Explain heart sounds.
Answer:
During each cardiac cycle, two prominent sounds are produced which can be easily heard through a stethoscope. The first heart sound (lub) is associated with the closure of the tricuspid and bicuspid valves, whereas the second heart sound (dup) is associated with the closure of the semilunar valves. These sounds are of clinical diagnostic significance.

Question 14.
Draw a standard ECG and explain the different segments in it.
Answer:
Electrocardiograph (ECG): ECG is a graphical representation of the electrical activity of the heart during a cardiac cycle. A patient is connected to the machine with three electrical leads (one to each wrist and to the left ankle) that continuously monitor the heart activity. For a detailed evaluation of the heart’s function, multiple leads are attached to the chest region.

Each peak in, the ECG is identified with a letter from P to T that corresponds to a specific electrical activity of the heart. The P-wave represents the electrical excitation (or depolarization) of the atria, which leads to the contraction of both the atria. The QRS complex represents the depolarization of the ventricles, which initiates the ventricular contraction. The contraction starts shortly after Q and marks the beginning of the systole.

• The T-wave represents the return of the ventricles from excited to normal state (repolarisation).
• The end of the T-wave marks the end of systole.
• Obviously, by counting the number of QRS complexes that occur in a given time period, one can determine the heartbeat rate of an individual.
• Since the ECGs obtained from different individuals have roughly the same shape for a given lead configuration, any deviation from this shape indicates a possible abnormality or disease.
• Hence, it is of a great clinical significance.

Punjab State Board PSEB 11th Class Biology Book Solutions Morphology of Flowering Plants Textbook Exercise Questions and Answers.

PSEB Solutions for Class 11 Biology Morphology of Flowering Plants

PSEB 11th Class Biology Guide Morphology of Flowering Plants Textbook Questions and Answers

Question 1.
What is meant by modification of root? What type of modification of root is found in the:
(a) Banyan tree
(b) Turnip
(c) Mangrove trees
Answer:
Modification of Root: Roots in some plants change their shape and
structure and become modified to perform functions, other than absorption and conduction of water and minerals. The roots are modified for water, absorption, support, storage of food and respiration.
(a) A banyan tree have hanging roots known as prop roots.
(b) The roots of turnip get modified to become swollen and store food.
(c) The roots of mangrove trees get modified to grow vertically upwards and help to get oxygen for respiration. These are known as pneumatophores.

Question 2.
Justify the following statements on the basis of external features:
(a) Underground parts of a plant are not always roots.
(b) Flower is a modified shoot.
Answer:
(a) Underground parts of a plant are not always roots, they are subterranean stems which do not have root hairs and root cap. Have terminal bud, nodes and internodes. Have leaves on the nodes.
Most of the underground stems such as sucker, rhizome, corm, tubers, bulb, etc., store food, form aerial shoots.
(b) Flower is a modified shoot because:

• It possess nodes and internodes.
• It may develop in the axil of small leaf-like structure called bract.
• Flowers get modified into bulbils or fleshy buds in some plants.
• Anatomically the pedicel and thalamus of a flower resemble that of stem.
• The vascular supply of different organs of flower resemble that of normal leaves.
• In the flower of Degeneria, the stamens are expanded like leaves and the carpels appear like folded leaves.

Question 3.
How is a pinnately compound leaf different from a palmately compound leaf?
Answer:
In pinnately compound leaf, the number of leaflets are present on a common axis, the rachis, which represents the midrib of the leaf as in neem. In case of a palmately compound leaf, the leaflets are attached at a common point, i e., at the tip of petiole as in silk cotton.

Question 4.
Explain with suitable examples the different types of phyllotaxy.
Answer:
Phyllotaxy is the pattern of arrangement of leaves on the stem or branch. This is usually of three types—alternate, opposite and whorled.

• In alternate phyllotaxy, a single leaf arises at each node in alternate manner, as in China rose, mustard and sunflower plants.
• In opposite type of phyllotaxy, a pair of leaves arise at each node and lie opposite to each other as in Calotropis and guava plants.
• If more than two leaves arise at each node and form, a whorl, it is called as whorled, as in Alstonia.

Question 5.
Define the following terms:
(i) Aestivation
(ii) Placentation
(iii) Actinomorphic
(iv) Zygomorphic
(v) Superior ovary
(vi) Perigynous flower
(vii) Epipetalous stamen
Answer:
(i) Aestivation: The mode of arrangement of sepals or petals in relation to one another in a flower bud is called aestivation.
(ii) Placentation: The pattern by which the ovules are attached in an ovary is called placentation.

(iii) Actinomorphic: A flower having radial symmetry. The parts of each whorl are similar in size and shape. The flower can be divided in two equal halves along more than one median longitudinal plane.

(iv) Zygomorphic: A flower having bilateral symmetry. The parts of one or more whorls are dissimilar. The flower can be divided into two equal halves in only one vertical plane.

(v) Superior ovary: The ovary is called superior when it is borne above the point attachment of perianth and stamens on the thalamus.

(vi) Perigynous flower: It is the condition in which gynoecium of a flower is situated in the centre and other parts of the flower are located on the rim of the thalamus almost at the same level.

(vii) Epipetalous stamen: Stamens adhere to the petals by their filaments so, appear to arise from them.

Question 6.
Differentiate between:
(a) Racemose and cymose inflorescence
(b) Fibrous root and adventitious root
(c) Apocarpous and syncarpous ovary
Answer:
(a) Differences between Racemose and Cymose Inflorescence.

Racemose	Cymose
1. This is further divided into (a) Raceme (b) Catkin (c) Spike (d) Spadix (e) Corymb (f) Umbel or capitutum	It is further divided into (a) Monochasial cyme (b) Dichasial Cyme (c) Polychasial Cyme.
2. Branches develop indefinitely and further branches arise laterally in acropetal manner.	The branches arise from terminal buds and stop growing after some time Lateral branches grow much vigorously and spread like a dome.

(b) Differences between Fibrous Root and Adventitious Root

Fibrous Root	Adventitious Root
In monocotyledonous plants, the primary root is short lived and is replaced by a latge number of roots. These roots originate from the base of the fibrous root system say, for example in wheat plants.	In some plants, say for example, in grass and banyan tree there are roots arising from parts of the plant other than the radicle. These are called adventitious roots.

(c) Differences between Apocarpous Ovaiy and Syncarpous Ovary

Apocarpous Ovary	Syncarpous Ovary
When more than one carpel is present, they may be free (as in lotus and rose) and are called apocarpous ovary.	They are termed syncarpous ovary when fused, as in mustard and tomato. After fertilisation, the ovules develop into seeds and like ovary matures into a fruit.

Question 7.
Draw the labelled diagram of the following:
(a) gram seed
(b) V.S. of maize seed
Answer:
(a) Gram Seed

(b) V.S. of maize seed

Question 8.
Describe modifications of stem with suitable examples. [NCERT]
Answer:
Modifications of stem are as follows:

• Tendrils help plants to climb on the support, e. g., Cucumber.
• Thorns are woody, pointed, straight structures to protect plants from browsing animals, e. g., Bougainvillea.
• The plants in arid regions modify their stems into flattened (Opuntia) or fleshy cylindrical (Euphorbia) structures. They contain chlorophyll and carry out photosynthesis.
• Underground stems of some plants such as grass and snawberry, etc., spread to new riches and when older parts die, new plants are formed.
• In Pistia and Eichhornia, a lateral branch with short internodes and each node bearing a rosette of leaves and a tuft of roots is found.
• Stolons or runners help in vegetative propagation in jasmine and grass, respectively.

Question 9.
Take one flower each of the families Fabaceae and Solanaceae and write its semi-technical description. Also draw their floral diagram after studying them.
Answer:
(i) Fabaceae: This family was earlier called Papilionoideae, a subfamily of family Leguminosae.
(a) Habit: Trees, shrubs, herbs, climbers, etc.
(b) Root System: Tap root system with root nodules, that harbour nitrogen fixing bacterium.
(c) Leaves: Leaves are alternate, simple or pinnately compound, pulvinate, and stipulate; venation reticulate.
(d) Inflorescence: Racemose usually, a raceme.
(e) Flowers: Bracteate, bracteolate, bisexual, zygomorphic, hypogynous, and pentamerous.
(f) Calyx: Five sepals, gamosepalous, irregular, odd sepal anterior (characteristic feature of the family) and valvate aestivation.

(g) Corolla: Corolla consists of five petals, polypetalous, characteristically papilionaceous, with an odd posterior large petal called standard or vexillum, a pair of lateral petals, called wing or alae and two anterior keel or carina, which enclose the essential organs; aestivation is vexillary.
(h) Androecium: Ten stamens, diadelphous, [(9) + 1] and anthers dithecous.
(i) Gynoecium: Ovary is superior, monocarpellary, unilocular with many ovules on marginal placenta; style single, curved or bent at right angles to the ovary.
(j) Fruits and Seeds: Characteristically a legume/pod and seeds are non-endospermic.
(k) Floral Formula:
(l) Economic Importance: Plants of this family yield pulses, edible oil, dye, fodder, fibres and wood; some yield products of medicinal value,

(ii) Solanaceae (Potato family)
(a) Habit: Plants are mostly herbs or shrubs or small trees; stem is erect, cylindrical, branched (cymose type); stem is underground in potato CSolarium tuberosum).
(b) Leaves: Simple, alternate, exstipulate with reticulate venation.
(c) Inflorescence: Axillary or extra-axillary cymose, or solitary.
(d) Flowers: Bisexual, actinomorphic, hypogynous and pentamerous.
(e) Calyx: Five sepals, gamosepalous, persistant and valvate aestivation.
(f) Corolla: Five petals, gamopetalous, valvate or imbricate, rotate/wheel-shaped.
(g) Androecium: Five stamens, epipetalous and alternating with the petals.
(h) Gynoecium: Bicarpellary, syncarpous, superior with many ovules on swollen axile placenta; carpels are obliquely placed.
(i) Fruits and Seeds: A berry (tomato and brinjal) or a capsule; seeds are endospermic.
(j) Floral Formula :
(k) Economic Importance: Many plants are used as source of food (vegetables), spice, medicines of fumigatory; some are ornamental plants.

Question 10.
Describe the various types of placentations found in flowering plants.
Answer:
Types of Placentations: The arrangement of ovules within the ovary is known as placentation. The placentations are of different types – marginal, axile, parietal, free central and basal.

Marginal placentation: In this placentation, the placenta forms a ridge along the ventral suture of the ovary and the ovules are borne on this ridge forming two rows as in pea.

Axile placentation: In this placentation, the placenta is axile and the ovules are attached to it in a multilocular ovary as in China rose, tomato, etc.

Parietal placentation: In this placentation, the ovules develop on the inner wall of the ovary or on peripheral part. Ovary is one chambered but it becomes two chambered due to the formation of a false septum known as replam, e.g., mustard.

Free central placentation: In this type of placentation, the ovules are present on the central axis of ovary and septa are absent as in Dianthus and primrose.

Basal placentation: In this placentation the placenta develops at the base of ovary and a single ovule is attached to it, as in sunflower.

Question 11.
What is a flower? Describe the parts of a typical angiospermic flower.
Answer:
Flower: It is a condensed modified reproductive shoot found in angiosperms. It often develops in the axile of a small leaf-like structure called bract. The stalk of the flower is called pedicel. The tip of the pedicel or the base of flower has a broad highly condensed multinodal region called thalamus.
A flower has following four floral structures:

• Calyx: It is made up of sepals. These are green in colour and help in photosynthesis.
• Corolla: It is the brightly coloured part containing petals.
• Androecium: It is the male reproductive part which consists of stamens. A stamen has a long filament and terminal anther. The anther produces the pollen grains.
• Gynoecium: It is the female reproductive part which consists of
  carpels. A carpel has three parts, i.e., style, stigma and ovary. The ovary bears the ovules.

Question 12.
How do the various leaf modifications help plants?
Answer:
Leaf Modifications in Plants
(i) In some plants, the leaf and leaf parts get modified to form green, long, thin unbranched and sensitive thread-like structures called tendrils. The tendrils coil around the plant and provide support to the plant in climbing. Tendrils are present in pea, garden Nasturtium, Clematis, Smilax, etc.

(ii) In some plants, the leaves get modified to form curved stiff claw like hooks to help the plant in clinging to the support. Leaflet hooks are present in Bignonia.

(iii) In case of Acacia and Zizyphus, the leaves get modified to form vasculated, hard, stiff and pointed structures.

(iv) In case of Acacia longifolia, the expanded petiole gets modified and perform the function of photosynthesis in absence of lamina.

(v) In plants such as Nepenthes, the lamina is modified to form large pitcher. It is used for storing water and for digesting insect protein.

(vi) In case of Utricularia, the leaf segments are modified into small bladders, to trap small animals.

Question 13.
Define the term inflorescence. Explain the basis for the different types of inflorescence in flowering plants.
Answer:
The arrangement of flowers on the floral axis is termed as inflorescence. Depending on whether the apex gets converted into a flower or continues to grow, two major types of inflorescences are defined – racemose and cymose. In racemose type of inflorescence the main axis continues to grow, the flowers are borne laterally in an acropetal succession.

In cymose type of inflorescence the main axis terminates in a flower, hence is limited in growth. The flowers are borne in a basipetal order, as depicted in figure.

Question 14.
Write the floral formula of a actinomorphic, bisexual, hypogynous flower with five united sepals, five free petals, five f free stamens and two united carpels with superior ovary and axile placentation.
Answer:
Floral formula

Question 15.
Describe the arrangement of floral members in relation to their ‘ insertion on thalamus.
Answer:
A flower is a condensed specialised reproductive shoot found in angiosperms. The stalk of the flower is known as pedicel. The tip of the pedicel or the base of the flower has a broad highly condensed multinodal region called thalamus. The floral parts of a flower are present on the thalamus. Starting from below they are green sepals or calyx, coloured petals or corolla, stamens or androecium and carpels or gynoecium.

Punjab State Board PSEB 11th Class Physics Book Solutions Chapter 15 Waves Textbook Exercise Questions and Answers.

PSEB Solutions for Class 11 Physics Chapter 15 Waves

PSEB 11th Class Physics Guide Waves Textbook Questions and Answers

Question 1.
A string of mass 50 kg is under a tension of 200 N. The length of the stretched string is 20.0 m. If the transverse jerk is struck at one end of the string, how long does the disturbance take to reach the other end?
Solution:
Mass of the string, M = 2.50 kg
Tension in the string, T = 200 N
Length of the string, l = 20.0 m
Mass per unit length, µ = \(\frac{M}{l}=\frac{2.50}{20} \) = 0.125 kg m^-1
The velocity (υ) of the transverse wave in the string is given by the relation:
υ = \(\sqrt{\frac{T}{\mu}}=\sqrt{\frac{200}{0.125}}=\sqrt{1600} \) = 40 m/s

∴ Time taken by the disturbance to reach the other end, t = \(\frac{l}{v}=\frac{20}{40}\) = 0.5 s

Question 2.
A stone dropped from the top of a tower of height 300 m high splashes into the water of a pond near the base of the tower. When is the splash heard at the top given that the speed of sound in air is 340ms^-1?(g=9.8ms^-2)
Solution:
Height of the tower, s = 300 m
The initial velocity of the stone, µ = 0
Acceleration, a = g = 9.8 m/s²
Speed of sound in air = 340 rn/s

The time (t₁) taken by the stone to strike the water in the pond can be calculated using the second equation of motion, as:
s=ut₁+\(\frac{1}{2}\) gt₁²
300 = 0+ \( \frac{1}{2}\) × 9.8 × t₁²
∴ t₁ =\( \sqrt{\frac{300 \times 2}{9.8}}\) = 7.82 s
Time taken by the sound to reach the top of the tower,
t₂ =\(\frac{300}{340}\) = 0.88 s
Therefore, the time after which the splash is heard, t = t₁ +t₂
= 7.82+0.88= 8.7 s .

Question 3.
A steel wire has a length of 12.0 m and a mass of 2.10 kg. What should be the tension in the wire so that the speed of a transverse wave on the wire equals the speed of sound in dry air at 20°C = 343 ms^-1.
Solution:
Length of the steel wire, l = 12 m
Mass of the steel wire, m = 2.10 kg
Velocity of the transverse wave, ν = 343 m/s
Mass per unit length, µ =\(\frac{m}{l}=\frac{2.10}{12}\) = 0.175 kg m^-1
For tension T, velocity of the transverse wave can be obtained using the relation:
υ = \(\sqrt{\frac{T}{\mu}}\)
∴ T = υ²µ = (343)² × 0.175 = 20588.575 ≈ 2.06 ×10⁴ N

Question 4.
Use the formula υ = \(\sqrt{\frac{\gamma \boldsymbol{P}}{\rho}}\) to explain why the speed of sound in
air
(a) is independent of pressure,
(b) increases with temperature,
(c) increases with humidity.
Solution:
(a) Given the relation:
υ = \( \sqrt{\frac{\gamma \boldsymbol{P}}{\rho}} \) ……………………………. (i)
Density, ρ = \(\frac{\text { Mass }}{\text { Volume }}=\frac{M}{V}\)
where, M = Molecular weight of the gas; V = Volume of the gas
Hence, equation ‘(i) reduces to:
υ = \(\sqrt{\frac{\gamma P V}{M}}\) …………………………………………. (ii)
Now, from the ideal gas equation for n = 1 :
PV = RT
For constant T, PV = Constant
Since both M and γ are constants, υ = Constant
Hence, at a constant temperature, the speed of sound in a gaseous
medium is independent of the change in the pressure of the gas.

(b) Given the relation:
υ = \( \sqrt{\frac{\gamma \boldsymbol{P}}{\rho}} \) …………………………………. (i)
For one mole of an ideal gas, the gas equation can be written as:
PV = RT
P = \(\frac{R T}{V}\)

Substituting equation (ii) in equation (i), we get:
υ = \(\sqrt{\frac{\gamma R T}{V \rho}}=\sqrt{\frac{\gamma R T}{M}}\) ……………………………… (iii)
where, M = mass = ρV is a constant; γ and R are also constants We conclude from equation (iii) that v ∝ \(\sqrt{T}\) .
Hence, the speed of sound in a gas is directly proportional to the square root of the temperature of the gaseous medium, i. e., the speed of the sound increases with an increase in the temperature of the gaseous medium and vice versa.

(c) Let υ_m and υ_d be the speeds of sound in moist air and dry air respectively.
Let ρ_m and ρ_d be the densities of moist air and dry air respectively.
Take the relation:
υ = \(\sqrt{\frac{\gamma P}{\rho}}\)
Hence, the speed of sound in moist air is:
υ_m = \(\sqrt{\frac{\gamma P}{\rho_{m}}}\) ………………………….. (i)
And the speed of sound in dry air is:
υ_d = \(\sqrt{\frac{Y P}{P_{d}}}\) ………………………………… (ii)

On dividing equations (i) and (ii), we get:
\(\frac{v_{m}}{v_{d}} \)
On dividing equations (i) and (ii), we get:
\(\frac{v_{m}}{v_{d}}=\sqrt{\frac{\gamma P}{\rho_{m}}} \times \frac{\rho_{d}}{\gamma P}=\sqrt{\frac{\rho_{d}}{\rho_{m}}} \)
However, the presence of water vapour reduces the density of air, i.e.,
ρ_d>ρ_m
∴ υ_m>υ_d
Hence the speed of sound in moist air greater than it is in dry air.
Thus, in a gaseous medium, the speed of sound increase with humidity.

Question 5.
You have learnt that a travelling wave in one dimension is represented by a function y = f(x, t) where x and t must appear
in the combination x-υt or x+υt, i.e., y=f(x±υt). Is the converse true? Examine if the following functions for y can
possibly represent a traveIliig wave:
(a) (x—υt)² (b)log \(\left[\frac{x+v t}{x_{0}}\right] \) (c) \(\frac{1}{(x+v t)}\)
Solution:
No, the converse of the given statement is not true. The essential requirement for a function to represent a travelling wave is that it should
remain finite for all values of x and t.

(a) Does not represent a wave
Explanation :
For x = 0 and t = 0, the function (x – υt)² becomes 0.
Hence, for x = 0 and t = 0, the function represents a point and not a wave,

(b) Represents a wave Explanation:
For x = 0 and t = 0, the function log \(\left(\frac{x+v t}{x_{0}}\right)\) = log 0 = ∞
Since the function does not converge to a finite value for x = 0 and t = 0, it represents a travelling wave.

(c) Does not represents a wave
Explanation :
For x = 0 and t = 0, the function
\(\frac{1}{x+v t}\) = log \(\frac{1}{0} \) = ∞
Since the function does not converge to a finite value for x = 0 and t = 0, it does not represent a travelling wave.

Question 6.
A bat emits the ultrasonic sound of frequency 1000 kHz in air. If the sound meets a water surface, what is the wavelength of (a) the reflected sound, (b) the transmitted sound? Speed of sound in air is 340 ms^-1 and in water 1486 ms^-1.
Solution:
(a) Frequency of the ultrasonic sound, υ = 1000 kHz = 10⁶ Hz
Speed of sound in water, υ_a = 340 m/s
The wavelength (λ_r)of the transmitted sound is given as:
λ_r = \(\frac{v}{v}=\frac{340}{10^{6}}\) = 3.4 × 10^-4 m

(b) Frequency of the ultrasonic sound, v = 1000 kHz = 10⁶ Hz
Speed of sound in water, υ_w, =1486 m/s
The wavelength of the transmitted sound is given as:
λ_t= \(\frac{1486}{10^{6}}\) = 1.49 × 10^-3 m

Question 7.
A hospital uses an ultrasonic scanner to locate tumours in a tissue. What is the wavelength of sound in the tissue in which the speed of sound is 1.7 km s^-1? The operating frequency of the scanner is 4.2 MHz.
Solution:
Speed of sound in the tissue, υ = 1.7 km/s = 1.7 x 10 3 m/s
Operating frequency of the scanner, v = 4.2 MHz = 4.2 x 10⁶ Hz
The wavelength of sound in the tissue is given as:
λ = \(\frac{v}{v}=\frac{1.7 \times 10^{3}}{4.2 \times 10^{6}}\) = 4.1 x 10^-4m.

Question 8.
A transverse harmonic wave on a string is described by y(x, t) = 3.0sin(36t+0.018x+\(\frac{\pi}{4}\))
where x and y are in cm and t in s. The positive direction of x is from left to right.
Is this a travelling wave or a stationary wave?
(a) If it is travelling, what are the speed and direction of its propagation?
(b) What are its amplitude and frequency?
(c) What is the initial phase at the origin?
(d) What is the least distance between two successive crests in the wave?
Solution:
(a) Yes.
The equation of a progressive wave travelling from right to left is given by the displacement function:
y(x,t) = a sin(ωt + kx + Φ) ……………………………………. (i)
The given equation is
y(x, t) = 3.0 sin( 36t +0.018x+\(\frac{\pi}{4}\)) …………………………………. (ii)
On comparing both the equations, we find that equation (ii) represents a travelling wave, propagating from right to left.
Now, using equations (i) and (ii), we can write:
ω = 36 rad/s and k = 0.018 cm^-1
We know that
v = \(\frac{\omega}{2 \pi}\) and λ = \(\frac{2 \pi}{k}\)
Also,
υ = vλ
∴ υ = \(\left(\frac{\omega}{2 \pi}\right) \times\left(\frac{2 \pi}{k}\right)\) = \(\frac{\omega}{k}=\frac{36}{0.018} \) = 2000 cm/s = 20 m/s
Hence, the speed of the given travelling wave is 20 m/s.

(b) Amplitude of the given wave, a =3 cm (Given)
Frequency of the given wave:
v = \(\frac{\omega}{2 \pi}=\frac{36}{2 \times 3.14}\) = 5.73 Hz

(c) On comparing çquations (i) and (ii), we find that the initial phase angle, Φ = \(\frac{\pi}{4}\)

(d) The distance between two successive crests or troughs is equal to the
wavelength of the wave.
Wavelength is given by the relation:
k= \(\frac{2 \pi}{\lambda}\)
∴ λ = \(\frac{2 \pi}{k}=\frac{2 \times 3.14}{0.018}\) = 348.89 cm = 3.49 m.

Question 9.
For the wave described in question 8, plot the displacement (y) versus (t) graphs for s =0,2 and 4 çm. What are the shapes of these graphs? In which aspects does the oscillatory motion in travelling wave differ from one point to another: amplitude, frequency or phase?
Solution:
All the waves have different phases. The given transverse harmonic wave is
y(x,t) = 3.0 sin (36t+0.018x + \(\frac{\pi}{4}\))
For x = 0, the equation reduces to
y(0,t) = 3.0 sin (36t+\(\frac{\pi}{4}\)) ………………………….. (i)
Also,
ω = \(\frac{2 \pi}{T}\) = 36 rad/s
∴ T = \(\frac{2 \pi}{\omega}=\frac{2 \pi}{36} \) = \(\frac{\pi}{18}\) s
For different values of t, we calculate y using eq. (i). These values are tabulated below

On plotting y versus t graph, we obtain a sinusoidal curve as shown in figure below.

Similar graphs are obtained for x = 2 cm and x = 4 cm.
The oscillatory motion in travelling wave differs from one point to another only in terms of phase. Amplitude and frequency of oscillatory motion remain the same in all the three cases.

Question 10.
For the travelling harmonic wave
y(x, t) = 2.0 cos 2π (10t – 0.0080x +0.35)
where, x and y are in cm and t in s. Calculate the phase difference between oscillatory motion of two points separated by a distance of
(a) 4m
(b) 0.5m
(c) \(\frac{\lambda}{2}\)
(d) \(\frac{3 \lambda}{4}\)
Solution:
Equation for a travelling harmonic wave is given as
y{x,t) = 2.0 cos 2π(10t -0.0080x +0.35)
= = 2.0 cos (20πt – 0.016πx+0.70π)

where, propagation constant, k = 0.0160π
Amplitude, a = 2 cm
Angular frequency, ω = 20 π rad/s
Phase difference is given by the relation:
Φ =kx=\(\frac{2 \pi}{\lambda} \)

(a)
For x=4m=400cm
Φ =0.016 π × 400 =6.4 π rad

(b) For 0.5 m=50cm
Φ = 0.1016 π × 50 = 0.8 π rad

(c) For x= \(\frac{\lambda}{2}\)
Φ = \(\frac{2 \pi}{\lambda} \times \frac{\lambda}{2}\) = π rad

(d) For x= \(\frac{3 \lambda}{4}\)
Φ = \(\frac{2 \pi}{\lambda} \times \frac{3 \lambda}{4} \) = 1.5π rad

Question 11.
The transverse displacement of a string (clamped at its both ends) is given by y(x,t) = 0.06 sin \(\frac{\mathbf{2} \pi}{\mathbf{3}}\) x cos (120πt) where x and y are in m and t in s. The length of the string is 1.5 m and its mass is 3.0 × 10^-2 Kg Answer the following
(a) Does the function represents a travelling wave or a stationary wave?
(b) Interpret the wave as a superposition of two waves travelling in opposite directions. What is the wavelength,
frequency, and speed of each wave?
(c) Determine the tension in the string.
Solution:
(a) The general equation representing a stationary wave is given by the displacement function:
y(x,t) = 2asinkxcosωt
This equation is similar to the given equation:
y(x,t)= 0.06 sin\(\left(\frac{2 \pi}{3} x\right)\) cos (120πt)
Hence, the given function represents a stationary wave.

(b) A wave travelling along the positive x -direction is given as
y₁ =asin(ωt -kx)
The wave travelling along the negative x -direction is given as:
y₂ = -asin(ωt +kx)
The superposition of these two waves yields:
y= y₁+y₂ = asin(ωt -kx)-asin(ωt +kr)
= asin(ωt)cos(kx) – asin(kx)cos(ωt)- asin(ωt)cos(kx) – asin(kx)cos(ωt)
= -2asin(kx)cos(ωt)
= – 2asin \(\left(\frac{2 \pi}{\lambda} x\right)\)cos (2πcvt) …………………………….. (i)

The transverse displacement of the string is given as y(x,t) = 0.06 sin \(\left(\frac{2 \pi}{3} x\right)\)cos (120πt) ………………………………. (ii)
Comparing equations (i) and (ii), we have
\(\frac{2 \pi}{\lambda}=\frac{2 \pi}{3}\)
∴ Wavelength, λ = 3 m
it is given that
120 π =2πv
Frequency, ν =60 Hz
Wave speed, υ = vλ
=60 × 3=180 m/s

(c) The velocity of a transverse wave travelling in a string is given by the relation
υ = \(\sqrt{\frac{T}{\mu}} \) ………………………… (iii)
where, µ = Mass per unit length of the string = \(\frac{m}{l}=\frac{3.0}{1.5} \times 10^{-2}\)
=2 x 10^-2 kgm^-1
T = Tension in the string = T
From equation (iii), tension can be obtained as
T =ν²µ=(180)² x 2 x 10^-2 =648 N

Question 12.
(i) For the wave on a string described in question 11, do all the points on the string oscillate with the same
(a) frequency,
(b) phase, (c) amplitude? Explain your answers.
(ii) What is the amplitude of a point 0.375 m away from one end?
Solution:
(I) (a) Yes, except at the nodes; All the points on the string oscillate with the same frequency, except at
the nodes which have zero frequency.

(b) Yes, except at the nodes;
All the points in any vibrating loop have the same phase, except at the nodes.

(C) No;
All the points in any vibrating loop have different amplitudes of vibration.

(ii) The given equation is .
y(x,t) = 0.06 sin \(\left(\frac{2 \pi}{3} x\right)\)cos (120πt)
For x = 0.375m and t =0
Amplitude = Displacement 0.06sin \(\left(\frac{2 \pi}{3} x\right) \cos 0^{\circ}\)
= 0.06 sin \(\left(\frac{2 \pi}{3} \times 0.375\right) \times 1\)
= 0.06 sin(0.25π) = 0.06 sin\(\left(\frac{\pi}{4}\right)\)
= 0.06 x \(\frac{1}{\sqrt{2}}\) = 0.042 m

Question 13.
Given below are some functions of x and t to represent the displacement (transverse or longitudinal) of an elastic wave. State which of these represent (i) a travelling wave, (ii) a stationary wave or (iii) none at all:
(a) y = 2cos(3x)sin(10t)
(b) y = 2\(\sqrt{x-v t}\)
(c) y = 3sin(5x – 0.5t) + 4cos(5x – 0.5t)
(d) Y = cos x sin t + cos 2x sin 2t
Solution:
(a) The given equation represents a stationary wave because the harmonic terms kx and cot appear separately in the equation.

(b) The given equation does not contain any harmonic term. Therefore, it does not represent either a travelling wave or a stationary wave.’

(c) The given equation represents a travelling wave as the harmonic terms kx and cot are in the combination of kx – cot.

(d) The given equation represents a stationary wave because the harmonic terms kx and cot appear separately in the equation. This equation actually represents the superposition of two stationary waves.

Question 14.
A wire stretched between two rigid supports vibrates in its fundamental mode with a frequency of 45 Hz. The mass of the wire is 3.5 x10^-2 kg and its linear mass density is 40 x 10^-2 kgm^-1. What is
(a) the speed of a transverse wave on the string, and
(b) the tension in the string?
Solution:
Mass of the wire, m = 3.5×10 ^-2 kg
Linear mass density, μ = \(\frac{m}{l}\) = 4.0 × 10^-2 kg m^-1
Frequency of vibration, μ = 45 Hz
∴ Length of the wire, l = \(\frac{m}{\mu}=\frac{3.5 \times 10^{-2}}{4.0 \times 10^{-2}}\) =0.875 m
The wavelength of the stationary wave (λ,) is related to the length of the wire by the relation:
λ = \(\frac{2 l}{n}\)
where, n = Number of nodes in the wire For fundamental node, n = 1:
λ =2l
λ =2 x 0.875 = 1.75 m
(a) The speed of the transverse wave in the string is given as
υ = vλ = 45 x 1.75 = 78.75 m/s

(b) The tension produced in the string is given by the relation:
T =υ² μ
= (78.75)² x 4.0 x 10^-2 =248.06 N

Question 15.
A metre-long tube open at one end, with a movable piston at the other end, shows resonance with a fixed frequency source (a tuning fork of frequency 340 Hz)when the tube length is 25.5 cm or 79.3 cm. Estimate the speed of sound in air at the temperature of the experiment. The edge effects may be neglected.
Solution:
Frequency of the turning fork, v = 340 Hz
Since the given pipe is attached with a piston at one end, it will behave as a pipe with one end closed and the other end open, as shown in the given figure.

Such a system produces odd harmonics. The fundamental note in a closed pipe is given by the relation l₁ = \(\frac{\lambda}{4}\)
where, length of the pipe, = 25.5 cm = 0.255 m
λ = 4l₁ =4 x 0.255 = 1.02 m
The speed of sound is given by the relation:
υ = vλ = 340 x 1.02 = 346.8 m/s

Question 16.
A steel rod 100 cm long is clamped at its middle. The fundamental frequency of longitudinal vibrations of the rod is given to be 2.53 kHz. What is the speed of sound in steel?
Solution:
Length of the steel rod, l = 100 cm = lm
v Fundamental frequency of vibration, v = 2.53 kHz = 2.53 x 10³ Hz When the rod is plucked at its middle, an antinode (A) is formed at its centre, and nodes (N) are formed at its two ends, as shown in the given figure.

The distance between two successive nodes is \(\frac{\lambda}{2}\)
l = \( \frac{\lambda}{2}\)
λ=2l=2 x l=2m
The speed of sound in steel is given by the relation:
v =vλ = 2.53 x 10³ x 2
= 5.06 x 10 ³ m/s = 5.06 km/s

Question 17.
A pipe 20 cm long is closed at one end. Which harmonic mode of the pipe is resonantly excited by a 430 Hz source? Will the same source be in resonance with the pipe if both ends are open? (Speed of sound in air is 340 ms^-1).
Solution:
First (Fundamental); No
Length of the pipe, l = 20 cm = 0.2 m
Source frequency = n^th normal mode of frequency, v_n = 430 Hz Speed of sound, ν = 340 m/s
In a closed pipe, n^th the rth normal mode of frequency is given by the relation
v_n = (2n -1) \(\frac{v}{4 l}\) ; n is an integer = 0,1,2,3 ………………
430 = (2n -1) \(\frac{340}{4 \times 0.2} \)
2n-1 = \(\frac{430 \times 4 \times 0.2}{340}\) = 1.01
2n =1.01+1
2n = 2.01
n ≈ 1

Hence, the first mode of vibration frequency is resonantly excited by the given source.
In a pipe open at both ends, the n^th mode of vibration frequency is given by the relation:
v_n = \(\frac{n v}{2 l}\)
n = \(\frac{2 l v_{n}}{v}=\frac{2 \times 0.2 \times 430}{340}\) = 0.5
Since the number of the mode of vibration (n) has to be an integer, the given source does not produce a resonant vibration in an open pipe.

Question 18.
Two sitar strings A and B playing the note ‘Ga’ are slightly out of tune and produce beats of frequency 6 Hz. The tension in string A is slightly reduced and the beat frequency is found to reduce to 3 Hz. If the original frequency of A is 324 Hz, what is the frequency of B?
Solution:
Frequency of string A, f_A = 324 Hz
Frequency of string B = f_B
Beat’s frequency, n = 6 Hz
Beat’s frequency is given as
n = |f_A ±f_B|
6 =324 ±f_B
f_B =330 Hz or 318 Hz

The frequency decreases with a decrease in the tension in a string. This is because the frequency is directly proportional to the square root of the tension. It is given as
v ∝ \(\sqrt{T}\)
Hence, the beat frequency cannot be 330 Hz.
∴ f_B= 318 Hz

Question 19.
Explain why (or how):
(a) In a sound wave, a displacement node is a pressure antinode and vice versa,
(b) Bats can ascertain distances, directions, nature, and sizes of the obstacles without any “eyes”,
(c) A violin note and sitar note may have the same frequency, yet we can distinguish between the two notes,
(d) Solids can support both longitudinal and transverse waves, but only longitudinal waves can propagate in gases, and
(e) The shape of a pulse gets distorted during propagation in a dispersive medium.
Solution:
(a) A node is a point where the amplitude of vibration is the minimum and pressure is the maximum. On the other hand, an antinode is a point where the amplitude of vibration is the maximum and pressure is the minimum.
Therefore, a displacement node is nothing but a pressure antinode and vice versa.

(b) Bats emit very high-frequency ultrasonic sound waves. These waves get reflected back toward them by obstacles. A bat receives a reflected wave (frequency) and estimates the distance, direction, nature, and size of an obstacle with the help of its brain senses.

(c) The overtones produced by a sitar and a violin, and the strengths of these overtones, are different. Hence, one can distinguish between the notes produced by a sitar and a violin even if they have the same frequency of vibration.

(d) Solids have shear modulus. They can sustain shearing stress. Since fluids do not have any definite shape, they yield to shearing stress. The propagation of a transverse wave is such that it produces shearing *’ stress in a medium. The propagation of such a wave is possible only in solids, and not in gases. ‘
Both solids and fluids have their respective bulk moduli. They can sustain compressive stress. Hence, longitudinal waves can propagate through solids and fluids.

(e) A pulse is actually a combination of waves having different wavelengths. These waves travel in a dispersive medium with different velocities, depending on the nature of the medium. This results in the distortion of the shape of a wave pulse.

Question 20.
A train, standing at the outer signal of a railway station blows a whistle of frequency 400 Hz in still air
(i) What is the frequency of the whistle for a platform observer when the train
(a) approaches the platform with a speed of 10 ms^-1,
(b) recedes from the platform with a speed of 10 ms^-1?
(ii) What is the speed of sound in each case? The speed of sound in still air can be taken as 340 ms ^-1.
Solution:
(i)
(a) Frequency of the whistle, ν = 400 Hz
Speed of the train, υ_T = 10 m/s
Speed of sound, υ = 340 m/s
The apparent frequency (v’) of the whisde as the train approaches the platform is given by the relation
υ’ = \(=\left(\frac{v}{v-v_{T}}\right) \mathrm{v}=\left(\frac{340}{340-10}\right) \times 400\) = 412.12 Hz
(b) The apparent frequency (v”) of the whistle as the train recedes from the platform is given by the relation
v” = \(\left(\frac{v}{v+v_{T}}\right) \mathrm{v}=\left(\frac{340}{340+10}\right) \times 400\) = 388.57 Hz

(ii) The apparent change in the frequency of sound is caused by the relative motions of the source and the observer. These relative motions produce no effect on the speed of sound. Therefore, the speed of sound in air in both the cases remains the same, i.e.,340 m/s.

Question 21.
A train standing in a station yard blows a whistle of frequency 400 Hz in still air. The wind starts blowing in the direction from the yard to the station with at a speed of 10 ms^-1. What are the frequency, wavelength, and speed of sound for an observer standing on the station’s platform? Is the situation exactly identical to the case when the air is still and the observer runs towards the yard at a speed of 10 ms^-1? The speed of sound in still air can be taken as 340 ms^-1.
Solution:
For the stationary observer:
Frequency of the sound produced by the whistle, v = 400 Hz
Speed of sound = 340 m/s
Velocity of the wind, ν = 10 m/s

As there is no relative motion between the source and the observer, the frequency of the sound heard .by the observer will be the same as that produced by the source, i. e., 400 Hz.
The wind is blowing toward the observer. Hence, the effective speed of the sound increases by 10 units, i.e.,
Effective speed of the sound, υ_e = 340 +10 = 350 m/s

The wavelength (λ) of the sound heard by the observer is given by the relation:
λ = \(\frac{v_{e}}{v}=\frac{350}{400}\) = 0.857 m

For the running observer:
Velocity of the observer, υ₀ = 10 m/s
The observer is moving toward the source. As a result of the
motions of the source and the observer, there is a change in (v’).
This is given by the relation:
υ’ = \(\left(\frac{v+v_{o}}{v}\right) v=\left(\frac{340+10}{340}\right) \times 400\) = 411.76 Hz
Since the air is still, the effective speed of sound = 340 + 0 = The source is at rest. Hence, the wavelength of the sound will i. e., λ remains 0.875 m
Hence, the given two situations are not exactly identical.

Additional Exercises

Question 22.
A travelling harmonic wave on a string is described by
y(x,t) = 7.5 sin (0.0050x + 12t+\(\frac{\pi}{4}\))
(a) What are the displacement and velocity of oscillation of a point at x = 1 cm, and t = 1 s? Is this velocity equal to the velocity of wave propagation?
(b) Locate the points of the string which have the same transverse displacements and velocity as the x = 1 cm point at t = 2 s, 5 s and 11 s.
Solution:
(a) The given harmonic wave is
y(x,t) = 7.5sin(0.0050x + 12t+\(\frac{\pi}{4}\))
For x = 1 cm and t = 1 s,
y(1, 1) = 7.5sin(0.0050x + 12t+\(\frac{\pi}{4}\))
= 7.5 sin (12.0050+\(\frac{\pi}{4}\)) = 7.5sinθ
where, 0 = 12.0050 + \(\frac{\pi}{4}\) = 12.0050 + \( \) = 12.79 rad 4
= \(\frac{180}{3.14} \times 12.79\) = 732.810
∴ y(1,1) = 7.5 sin (732.810) = 7.5sin (90 × 8 +12.81°) = 7.5 sin 12.81°
= 7.5 × 0.2217
= 1.6229 ≈ 1.663 cm
The velocity of the oscillation at a given point and time is given as

At x = 1 cm and t = 1 s
v = y(1, 1) = 90 cos(12.005 +\(\frac{\pi}{4}\))
= 90 cos (732.81 ° ) = 90 cos (90 x 8 +12.81 ° ) = 90cos(12.81°) = 90 x 0.975 = 87.75 cm/s

Now, the equation of a propagating wave is given by
y(x,t) = asin(kx +ωt +Φ)
where, k = \(\frac{2 \pi}{\lambda} \)
∴ λ = \(\frac{2 \pi}{k}\)
And ω = 2πv
∴ v = \(\frac{\omega}{2 \pi}\)
Speed, υ = vλ = \(\frac{\omega}{k}\)
where, ω = 12 rad/s
k = 0.0050 cm-1
∴ v = \(\frac{12}{0.0050}\) = 2400 cm/s
Hence, the velocity of the wave oscillation at x = 1 cm and t = 1 s is not equal to the velocity of the wave propagation.

(b) Propagation constant is related to wavelength as:
k = \(\frac{2 \pi}{\lambda}\)
∴ λ = \(\frac{2 \pi}{k}=\frac{2 \times 3.14}{0.0050}\) = 1256 cm = 12.56 cm
Therefore, all the points at distances nλ{n = ±1,±2… and so on), i.e., ±12.56 m, + 25.12m, … and so on for x =1 cm, will have the same displacement as the x = 1 cm points at t = 2 s, 5 s and 11s.

Question 23.
A narrow sound pulse (for example, a short pip by a whistle) is sent across a medium,
(a) Does the pulse have a definite
(i) frequency,
(ii) wavelength,
(iii) speed of propagation?
(b) If the pulse rate is 1 after every 20 s, (that is the whistle is blown for a split of second after every 20 s), is the frequency of the note
produced by the whistle equal to \(\frac{1}{20}\) or 0.05 Hz?
Solution:
(a) (i) No;
(ii) No;
(iii) Yes;
Explanation:
The narrow sound pulse does not have a fixed wavelength or frequency. However, the speed of the sound pulse remains the same, which is equal to the speed of sound in that medium.
(b) No;
The short pip produced after every 20 s does not mean that the frequency of the whistle is \(\frac{1}{20}\) or 0.05 Hz. It means that 0.05 Hz is the frequency of the repetition of the pip of the whisde.

Question 24.
One end of a long string of linear mass density 8.0 x 10-3 kg m-1 is connected to an electrically driven tuning fork of frequency 256 Hz. The other end passes over a pulley and is tied to a pan containing a mass of 90 kg. The pulley end absorbs all the incoming energy so that reflected waves at this end have negligible amplitude.

At t = 0, the left end (fork end) of the string x = 0 has zero transverse displacement (y = 0) and is moving along positive y-direction. The amplitude of the wave is 5.0 cm. Write down the transverse displacement y as a function of x and t that describes the wave on the string.
Solution:
The equation of a Gravelling wave propagating along the positive y-direction is given by the displacement equation
y(x, t) = a sin (cot – kx) ………………………. (i)
Linear mass density, μ = 8.0 x 10 -3 kg m-1
Frequency of the tuning fork, v = 256 Hz
The amplitude of the wave, a = 5.0 cm = 0.05 m ……………………………. (ii)
Mass of the pan, m = 90 kg
Tension in the string, T = mg = 90 x 9.8 = 882 N

The velocity of the transverse wave υ, is given by the relation:
υ = \(=\sqrt{\frac{T}{\mu}}=\sqrt{\frac{882}{8.0 \times 10^{-3}}}\) = 332 m/s
Angular Frequency, ω = 2πv
= 2 x 3.14 x 256
= 1607.68 = 16 x 103 rad/s ………………………….. (iii)
Wavelength λ = \(\frac{v}{v}=\frac{332}{256}\)m
∴ propagation constant, k = \(\frac{2 \pi}{\lambda}=\frac{2 \times 3.14}{\frac{332}{256}}\) = 4.84 m-1 ……….. (iv)
Substituting the values from equations (ii), (iii), and (iv) in equation (i), we get the displacement equation:
y(x,t) = 0.05sin(1.6 x 103t -4.84 x)
where x and y are in and t in s.

Question 25.
A SONAR system fixed in a submarine operates at a frequency 40.0 kHz. An enemy submarine moves towards the SONAR with a speed of 360 km h-1. What is the frequency of sound reflected by the submarine? Take the speed of sound in water to be 1450 ms-1.
Solution:
Operating frequency of the SONAR system, v = 40 kHz
Speed of the enemy submarine, ve = 360 km/h = 100 m/s
Speed of sound in water, v = 1450 m/s
The source is at rest and the observer (enemy submarine) is moving toward it.

Hence, the apparent frequency (v’) received and reflected by the submarine is given by the relation:
The frequency (v”) received by the enemy submarine is given by the relation
v’ = \( =\left(\frac{v+v_{e}}{v}\right) v=\left(\frac{1450+100}{1450}\right) \times 40\) = 42.76 kHz
The frequency (V”) received by the enemy submarine is given by the relation
v” = \(\left(\frac{v}{v-v_{s}}\right) v^{\prime}\)
Where vs = 100 m/s
∴ v” = \(\left(\frac{1450}{1450-100}\right) \times 42.76 \) = 45.93 kHz

Question 28.
Earthquakes generate sound waves inside the Earth. Unlike a gas, the Earth can experience both transverse (S) and
longitudinal (P) sound waves. Typically the speed of S wave is about 4.0kms-1 , and that of P wave is 8.0 kms1. A
seismograph records P and S waves from an Earthquake. The first P wave arrives 4 min before the first S wave. Assuming the waves travel in straight line, at what distance does the Earthquake occur?
Solution:
Let νs and vp, be the velocities of S and P waves respectively.
Let L be the distance between the epicentre and the seismograph.
We have
L = νstsub>s ……………………….. (i)
L = νptsub>p …………………………(ii)

where ts and tp are the respective times taken by the S and P waves to
reach the seismograph from the epicentre
It is given that
νp =8km/s
νs =4km/s

From equations (i) and (ii), we have
υsts = υptp
4ts = 8tp
ts = 2tp …………………………..(iii)
It is also given that
ts – tp =4 min=240s
2tp-tp= 240
tp = 240
and = 2×240 =840 s
From equation (ii), we get

L =8×240=1920 km
Hence, the Earthquake occurs at a distance of 1920 km from the seismograph.

Question 27.
A bat is flitting about in a cave, navigating via ultrasonic beeps. Assume that the sound emission frequency of the bat is 40 kHz. During one fast swoop directly toward a flat wall surface, the bat is moving at 0.03 times the speed of sound hi air. What frequency does the bat hear reflected off the wall?
Solution:
Ultrasonic beep frequency emitted by the bat, ν = 40 kHz
The velocity of the bat, νb = 0.03 ν
where, ν = velocity of sound in air
The appartment frequency of the sound strìking the wall is given as
v’ = \(\left(\frac{v}{v-v_{b}}\right) v=\left(\frac{v}{v-0.03 v}\right) \times 40 \) = \(\frac{40}{0.97}\) kHz
This frequency is reflected by the stationary wall (νs = 0) toward the bat.
The frequency (ν”) of the received sound is given by the relation:
ν” = \(\left(\frac{v+v_{b}}{v}\right) \mathrm{v}^{\prime}=\left(\frac{v+0.03 v}{v}\right) \times \frac{40}{0.97}=\frac{1.03 \times 40}{0.97}\) = 42.47 kHz

Punjab State Board PSEB 11th Class Physics Book Solutions Chapter 14 Oscillations Textbook Exercise Questions and Answers.

PSEB Solutions for Class 11 Physics Chapter 14 Oscillations

PSEB 11th Class Physics Guide Oscillations Textbook Questions and Answers

Question 1.
Which of the following examples represent periodic motion?
(a) A swimmer completing one (return) trip from one bank of a river to the other and back.
(b) A freely suspended bar magnet displaced from its N-S direction and released.
(c) A hydrogen molecule rotating about its center of mass.
(d) An arrow released from a bow.
Solution:
(b) and (c)
Explanations :
(a) The swimmer’s motion is not periodic. The motion of the swimmer between the banks of a river is back and forth. However, it does not have a definite period. This is because the time taken by the swimmer during his back and forth journey may not be the same.

(b) The motion of a freely-suspended magnet, if displaced from its N-S direction and released, is periodic. This is because the magnet oscillates about its position with a definite period of time.

(c) When a hydrogen molecule rotates about its center of mass, it comes to the same position again and again after an equal interval of time. Such motion is periodic.

(d) An arrow released from a bow moves only in the forward direction. It does not come backward. Hence, this motion is not a periodic.

Question 2.
Which of the following examples represent (nearly) simple harmonic motion and which represent periodic but not simple harmonic motion?
(a) the rotation of earth about its axis.
(b) motion of an oscillating mercury column in a 17-tube.
(c) motion of a ball bearing inside a smooth curved bowl, when released from a point slightly above the lowermost point.
(d) general vibrations of a polyatomic molecule about its equilibrium position.
Solution:
(b) and (c) are SHMs; (a) and (d) are periodic, but not SHMs
Explanations :
(a) During its rotation about its axis, earth comes to the same position again and again in equal intervals of time. Hence, it is a periodic motion. However, this motion is not simple harmonic. This is because earth does not have a to and fro motion about its axis.

(b) An oscillating mercury column in a [/-tube is simple harmonic. This is because the mercury moves to and fro on the same path, about the fixed position, with a certain period of time.

(c) The ball moves to and fro about the lowermost point of the bowl when released. Also, the ball comes back to its initial position in the same period of time, again and again. Hence, its motion is periodic as well as simple harmonic.

(d) A polyatomic molecule has many natural frequencies of oscillation. Its vibration is the superposition of individual simple harmonic motions of a number of different molecules. Hence, it is not simple harmonic, but periodic.

Question 3.
Figure depicts four x-t plots for linear motion of a particle. Which of the plots represents periodic motion? What is the period of motion (in case of periodic motion)?
(a)

(b)

(c)

(d)

Solution:
(b) and (d) are periodic
Explanation :
(a) It is not a periodic motion. This represents a unidirectional, linear uniform motion. There is no repetition of motion in this case.
(b) In this case, the motion of the particle repeats itself after 2 s. Hence, it is a periodic motion, having a period of 2 s.
(c) It is not a periodic motion. This is because the particle repeats the motion in one position only. For a periodic motion, the entire motion of the particle must be repeated in equal intervals of time. In this case, the motion of the particle repeats itself after 2 s. Hence, it is a periodic motion, having a period of 2 s.

Question 4.
Which of the following functions of time represent (a) simple ‘ harmonic, (b) periodic but not simple harmonic, and (c) non¬periodic motion? Give period for each case of periodic motion (a is any positive constant):
(a) sin ωt – cos ωt
(b) sin ³ωt
(c) 3cos(\(\pi / 4 \) -2ωt)
(d) cos ωt +cos 3 ωt+cos 5ωt
(e) exp(-ω²t²)
(f) 1+ ωt+ω^2t
Solution:
(a) SHM
The given function is:

This function represents SHM as it can be written in the form: a sin (ωt +Φ)
Its period is: \(\frac{2 \pi}{\omega}\)

(b) Periodic, but not SHM The given function is:
sin³ ωt = \(\frac{1}{4}\) [3sinωt -sin3ωt] (∵ sin3θ = 3sinθ – 4sin³ θ)
The terms sin cot and sin 3 ωt individually represent simple harmonic motion (SHM). However, the superposition of two SHM is periodic and not simple harmonic.

Period of\(\frac{3}{4}\)sin ωt = \(\frac{2 \pi}{\omega}\) = T
Period of\(\frac{1}{4}\)sin3ωt = \(\frac{2 \pi}{3 \omega}\) = T’ = \(\frac{T}{3}\)
Thus, period of the combination
= Minimum time after which the combined function repeats
= LCM of T and \(\frac{T}{3}\) = T
Its period is 2 \(\pi / \omega\)

(c) SHM
The given function is:
3 cos \(\left[\frac{\pi}{4}-2 \omega t\right]\) = 3 cos \(\left[2 \omega t-\frac{\pi}{4}\right]\)
This function represents simple harmonic motion because it can be written in the form:
acos(ωt +Φ)

Its period is :
\(\frac{2 \pi}{2 \omega}=\frac{\pi}{\omega}\)

(d) Periodic, but not SHM
The given function is cosωt +cos3ωt +cos5ωt. Each individual cosme function represents SHM. However, the superposition of threc simple harmonic motions is periodic, but not simple harmonic.

cosωt represents SHM with period = \(\frac{2 \pi}{\omega}\) T (say)
cos 3ωt represents SHM with period = \(\frac{2 \pi}{3 \omega}=\frac{T}{3}\)
cos 5ωt represents SHM with period = \(\frac{2 \pi}{5 \omega}=\frac{T}{5}\)
The minimum time after which the combined function repeats its value is T. Hence, the given function represents periodic function but not SHM, with period T.

(e) Non-periodic motion: .
The given function exp(- ω²t²) is an exponential function. Exponential functions do not repeat themselves. Therefore, it is a non-periodic motion.

(f) Non-periodic motion
The given function is 1+ ωt + ω²t²
Here no repetition of values. Hence, it represents non-periodic motion.

Question 5.
A particle Is in linear simple harmonic motion between two points, A and B, 10 cm apart Take the direction from A to B as the positive direction and give the signs of velocity, acceleration and force on the particle when It Is
(a) at the end A,
(b) at the end B,
(c) at the midpoint of AB going towards A,
(d) at 2 cm away from B going towards A,
(e) at 3 cm away from A going towards B, and
(f) at 4 cm away from B going towards A.
Solution:
The given situation is shown in the following figure. Points A and B are the two endpoints, with AB =10cm. 0 is the midpoint of the path.

A particle is in linear simple harmonic motion between the endpoints
(a) At the extreme point A, the particle is at rest momentarily. Hence, its velocity is zero at this point.
Its acceleration is positive as it is directed along with AO.
Force is also positive in this case as the particle is directed rightward.

(b) At the extreme point B, the particle is at rest momentarily. Hence, its velocity is zero at this point.
Its acceleration is negative as it is directed along B.
Force is also negative in this case as the particle is directed leftward.

(c)
The particle is extending a simple harmonic motion. O is the mean position of the particle. Its velocity at the mean position O is the maximum. The value for velocity is negative as the partide Is directed leftward. The acceleration and force of a particle executing SHM is zero at the mean position.

(d)
The particle is moving towards point O from the end B. This direction of motion is opposite to the conventional positive direction, which is from A to R. Hence, the particle’s velocity and acceleration, and the force on it are all negative.

(e)

The particle is moving towards point O from the end A. This direction of motion is from A to B, which is the conventional positive direction. Hence, the value for velocity, acceleration, and force are all positive.

(f)

This case is similar to the one given in (d).

Question 6.
Whi ch of the following relationships between the acceleration a and the displacement x of a particle involves simple harmonic motion?
(a) a=0.7x
(b) a=-200x²
(c) a= – 10 x (d) a=100x³
Solution:
A motion represents simple harmonic motion if it is governed by the force law:
F=-kx
ma’= -kx
∴ a = – \(\frac{k}{m}\) x

where F is the force
m is the mass (a constant for a body)
x is the displacement
a is the acceleration.
k is a constant
Among the given equations, only equation a = -10 x is written in the
above form with \( \frac{k}{m}\) =10. Hence, this relation represents SHM.

Question 7.
The motion of a particle executing simple harmonic motion is described by the displacement function,
x(t) = A cos (ωt+Φ)
If the initial (t = 0) position of the particle is 1 cm and its initial velocity is ω cm/s, what are its amplitude and initial phase angle? The angular frequency of the particle is πs^-1. If instead of the cosine function, we choose the sine function to describe the SHM:x = B sin(ωt + α), what are the amplitude and initial phase of the particle with the above initial conditions.
Solution:
Initially, at t = 0:
Displacement, x = 1 cm Initial velocity, ν = ω cm/s.
Angular frequency, ω = π rad s^-1
It is given that:
x(t) = Acos (ωt+Φ)
1 = Acos(ω x 0 +Φ) = AcosΦ
AcosΦ =1 ……………………………….. (i)

Velocity, ν = \(\frac{d x}{d t}\)
ω = -Aω sin(ωt +Φ)
1 = -Asin(ω x 0 +Φ) = -AsinΦ
Asin Φ = -1 ………………………… (ii)
Squaring and adding equations (i) and (ii), we get
A²(sin²Φ +cos²Φ) = 1+1
A² = 2
∴ A = \(\sqrt{2}\) cm

Dividing equation (ii) by equation (i), we get
tanΦ = -1

SHM is given as
x = Bsin(ωt + α)
Putting the given values in this equation, we get 1 =B sin (ωt + α)
B sin α =1 …………………………… (iii)

Velocity, ν = \(\frac{d x}{d t}\)
ω =(ωB cos (ωt + a)
1 =B cos (ω x 0+α ) = B cos α …………………………………… (iv)
Squaring and adding equations (iii) and (iv), we get
B² [sin²α +cos² α] =1+1
B² =2
B = \(\sqrt{2}\) cm
Dividing equation (iii) by equation (iv), we get

Question 8.
A spring balance has a scale that reads from 0 to 50kg. The length of the scale is 20 cm. A body suspended from this balance, when displaced and released, oscillates with a period of 0.6 s. What is the weight of the body?
Solution:
Maximum mass that the scale can read, M = 50 kg
Maximum displacement of the spring = Length of the scale, l = 20 cm = 0.2 m Time period, T =0.6s
Maximum force exerted on the spring, F = Mg where,
g = acceleration due to gravity = 9.8 m/s²
F = 50 × 9.8 = 490 N
∴ Spring constant , K = \(\frac{F}{l}=\frac{490}{0.2}\) = 2450Nm^-1

Mass m, is suspended from the balance,
Time period, T = \(2 \pi \sqrt{\frac{m}{k}}\)
∴ m = \(\left(\frac{T}{2 \pi}\right)^{2} \times k=\left(\frac{0.6}{2 \times 3.14}\right)^{2} \times 2450 \) = 22.36 kg
∴ Weight of the body = mg = 22.36 x 9.8 = 219.167N
Hence, the weight of the body is about 219 N.

Question 9.
Aspringhavingwith aspiring constant 1200Nm^-1 is mounted on a horizontal table as shown in figure. A mss of 3kg is attached to the free end of the spring. The mass is then pulled sideways to a distance of 2.0cm and released.

Determine (i) the frequency of oscillations,
(ii) maximum acceleration of the mass, and
(iii) the maximum speed of the mass.
Solution:
Spring constant, k = 1200 Nm^-1
mass,m = 3 Kg
Displacement,A = 2.0 cm = 0.02 m
(i) Frequency of oscillation y, is gyen by the relation
V = \(\frac{1}{T}=\frac{1}{2 \pi} \sqrt{\frac{k}{m}}\)
where, T is the time period
∴ v = \(\frac{1}{2 \times 3.14} \sqrt{\frac{1200}{3}}\) = 3.18 s^-1
Hence, the frequency of oscillations is 3.18 s^-1.

(ii) Maximum acceleration a is given by the relation:
a = ω²A
where,
ω = Angular frequency = \(\sqrt{\frac{k}{m}}\)
A = Maximum displacement
∴ a = \(\frac{k}{m} A=\frac{1200 \times 0.02}{3}\) = 8 ms^-2
Hence, the maximum acceleration of the mass is 8.0 ms²

(iii) Maximum speed, ν_max = Aω
= \(A \sqrt{\frac{k}{m}}=0.02 \times \sqrt{\frac{1200}{3}}\) = 0.4 m/s
Hence, the maximum speed of the mass is 0.4 m/s.

Question 10.
In question 9, let us take the position of mass when the spring is unstretched as x =0, and the direction from left to right as the positive direction of x-axis. Give x as a function of time t for the oscillating massif at the moment we start the stopwatch (t = 0), the mass is
(a) at the mean position,
(b) at the maximum stretched position, and
(c) at the maximum compressed position.
In what way do these functions for SHM differ from each other, in frequency, in amplitude br the initial phase?
Solution:
(a) The functions have the same frequency and amplitude, but different initial phases.
Distance travelled by the mass sideways, A = 2.0 cm
Force constant of the spring, k =1200 N m^-1
Mass, m =3kg
Angular frequency of oscillation,
ω = \(\sqrt{\frac{k}{m}}=\sqrt{\frac{1200}{3}} \) = \(\sqrt{400}\) = 20 rad s^-1
When the mass is at the mean position, initial phase is 0.
Displacement,
x = A sinωt = 2 sin20 t

(b) At the maximum stretched position, the mass is toward the extreme right. Hence, the initial phase is \(\frac{\pi}{2}\)
Displacement, x = Asin \(\left(\omega t+\frac{\pi}{2}\right)\)
=2sin\(\left(20 t+\frac{\pi}{2}\right)\)
= 2 cos 20t

(c) At the maximum compressed position, the mass is toward the extreme left. Hence, the initial phase is \(\frac{3 \pi}{2}\)
Displacement, x = A sin \(\left(\omega t+\frac{3 \pi}{2}\right)\)
= 2sin \(\left(20 t+\frac{3 \pi}{2}\right)\) = -2cos 20t
The functions have the same frequencyl \(\left(\frac{20}{2 \pi} \mathrm{Hz}\right)\) land amplitude (2cm),
but different initial phases \(\left(0, \frac{\pi}{2}, \frac{3 \pi}{2}\right)\)

Question 11.
Figures correspond to two circular motions. The radius of the circle, the period of revolution, the initial position, and the sense of revolution (i. e., clockwise or anti-clockwise) are indicated on each figure.

Obtain the corresponding simple harmonic motions of the x-projection of the radius vector of the revolving particle P, in each case.
solution:
Time period,T =2s
Amplitude, A = 3cm
At time, t = O, the radius vector OP makes an angle \(\frac{\pi}{2}\) with the positive x -axis, i.e., phase angle Φ = + \(\frac{\pi}{2}\)
Therefore, the equation of simple harmonic motion for the x —projection of OP, at time t, is given by the displacement equation

(b) Time period, T = 4s
Amplitude, a =2 m
At time t = 0, OP makes an angle ir with the x-axis, in the anticlockwise direction. Hence, phase angle, Φ = +π
Therefore, the equation of simple harmonic motion for the x -projection of OP, at time t, is given as

Question 12.
Plot the corresponding reference circle for each of the following simple harmonic motions. Indicate the initial (t = 0) position of the particle, the radius of the circle, and the angular speed of the rotating particle. For simplicity, the sense of rotation may be fixed to be anticlockwise in every case: (x is in cm and t is in s).
(a) x= – 2sin(3t+\(\pi / \mathbf{3}\) )
(b) x=cos (\(\pi / 6\) – t)
(c) x=3 sin (2πt + \(\pi / 4 \) )
(d) x=2 cos πt
Solution:
(a) x = -2 sin \(\left(3 t+\frac{\pi}{3}\right)=+2 \cos \left(3 t+\frac{\pi}{3}+\frac{\pi}{2}\right)=2 \cos \left(3 t+\frac{5 \pi}{6}\right) \)

If this equation is compared with the standard SHM equation,
x =A cos \(\left(\frac{2 \pi}{T} t+\phi\right)\) then we get
Amplitude, A = 2cm
Phase angle, Φ = \(\frac{5 \pi}{6}\) =150°
Angular velocity, ω = \(\frac{2 \pi}{T}\) =3 rad/sec
The motion of the particle can be pokted as shown in the following figure.

(b) x=cos \(\left(\frac{\pi}{6}-t\right)\) =cos \(\left(t-\frac{\pi}{6}\right)\)
If this equation is compared with the standard SHM equation,
x = A cos \(\left(\frac{2 \pi}{T} t+\phi\right)\) then we get
Amplitude, A = 1 cm
Phase angle, Φ = \(-\frac{\pi}{6} \) = – 30°
Angular velocity, ω = \(\frac{2 \pi}{T}\) =1 rad/s
The motion of the particle can be plotted as shown in the following figure.

(c) x =3sin \(\left(2 \pi t+\frac{\pi}{4}\right)\)

If this equation is compared with the standard SHM equation
x = Acos \(\left(\frac{2 \pi}{T} t+\phi\right)\) then we get
Amplitude, A = 3cm
Phase angle, Φ = \(-\frac{\pi}{4}\)
Angular velocity, ω = \(\frac{2 \pi}{T}\) = 2π rad/s
The motion of the particle can be plotted as shown in the following figure.

(d) x=2cosπt
If this equation is compared with the standard SHM equation,
x = A cos \(\left(\frac{2 \pi}{T} t+\phi\right) \) then we get
Amplitude, A = 2cm
Phase angle, Φ = 0
Angular velocity, ω = π rad/s
The motion of the particle can be plotted as shown in the following figure.

Question 13.
Figure (a) shows a spring of force constant k clamped rigidly at one end and a mass m attached to its free end. A force F applied at the free end stretches the spring. Figure (b) shows the same spring with both ends free and attached to a mass m at either end. Each end of the spring in figure (b) is stretched by the same force F.

(a) What is the minimum extension of the spring in the two cases?
(b) If the mass in Fig. (a) and the two masses in Fig. (b) are released, what is the period of oscillation in each case?
Solution:
(a) For the one block system:
When a force F, is applied to the free end of the spring, an extension l, is produced. For the maximum extension, it can be written as:
F=kl
where k is the spring constant
Hence, the maximum extension produced in the spring, l = \(\frac{F}{k}\)
For the two blocks system:
The displacement (x) produced in this case is:
x = \(\frac{l}{2}\)
Net force, F = +2kx =2k \(\frac{l}{2}\)
∴ l = \(\frac{F}{k}\)

(b) For the one blocks system:
For mass (m) of the block, force is written as
F = ma = m \(\frac{d^{2} x}{d t^{2}}\)
where, x is the displacement of the block in time t
∴ m \(\frac{d^{2} x}{d t^{2}}\) = -kx
It is negative because the direction of elastic force is opposite to the direction of displacement.

where, ω is angular frequency of the oscillation
∴ Time period of the oscillation,
T= \(\frac{2 \pi}{\omega}=\frac{2 \pi}{\sqrt{\frac{k}{m}}}=2 \pi \sqrt{\frac{m}{k}}\)

For the two blocks system:
F=m \(\frac{d^{2} x}{d t^{2}}\)
m \(\frac{d^{2} x}{d t^{2}}\) =-2kx

It is negative because the direction of elastic force is opposite to the direction of displacement.
\(\frac{d^{2} x}{d t^{2}}\) = \(-\left[\frac{2 k}{m}\right] x \) = – ω²x
where, Angular frequency, ω = \(\sqrt{\frac{2 k}{m}}\)
∴ Time period T = \(\frac{2 \pi}{\omega}=2 \pi \sqrt{\frac{m}{2 k}} \)

Question 14.
The piston in the cylinder head of a locomotive has a stroke (twice the amplitude) of 1.0 in. If the piston moves with simple harmonic motion with an angular frequency of 200 rad/min, what is its maximum speed?
Solution:
Angular frequency of the piston, ω = 200 rad/min.
Stroke =1.0 m
Amplitude, A = \(\frac{1.0}{2}\) = 0.5m
The maximum speed (ν_max) of the piston is given by the relation
ν_max =Aω = 200 x 0.5=100 m/min

Question 15.
The acceleration due to gravity on the surface of moon is 1.7 ms^-2. What is the time period of a simple pendulum on the surface of moon If Its time period on the surface of earth is 3.5 s? (gon the surface of earth is 9.8 ms^-2)
Solution:
Acceleration due to gravity on the surface of moon, g’ = 1.7m s^-2
Acceleration due to gravity on the surface of earth, g = 9.8 ms^-2
Time period of a simple pendulum on earth, T = 3.5 s
T= \(2 \pi \sqrt{\frac{l}{g}}\)

where l is the length of the pendulum
∴ l = \(\frac{T^{2}}{(2 \pi)^{2}} \times g=\frac{(3.5)^{2}}{4 \times(3.14)^{2}} \times 9.8 \mathrm{~m} \)

The length of the pendulum remains constant.
On Moon’s surface, time period,
T’ = \(2 \pi \sqrt{\frac{l}{g^{\prime}}}=2 \pi \sqrt{\frac{(3.5)^{2}}{\frac{4 \times(3.14)^{2}}{1.7}} \times 9.8} \) = 8.4 s
Hence, the time period of the simple pendulum on the surface of Moon is 8.4 s.

Question 16.
Answer the following questions:
(a) Time period of a particle in SHM depends on the force constant k and mass m of the particle:
T = \(2 \pi \sqrt{\frac{m}{k}}\) A simple pendulum executes SHM approximately. Why then is the time period of a pendulum independent of the mass of the pendulum?

(b) The motion of a simple pendulum is approximately simple harmonic for small-angle oscillations. For larger angles of oscillation a more involved analysis shows that T is greater than \(2 \pi \sqrt{\frac{l}{g}}\) Think of a qualitative argument to appreciate this result.

(c) A man with a wristwatch on his hand falls from the top of a tower. Does the watch give correct time during the free fall?

(d) What is the frequency of oscillation of a simple pendulum mounted in a cabin that is freely falling under gravity?
Solution :
(a) The time period of a simple pendulum, T = \(2 \pi \sqrt{\frac{m}{k}}\)
For a simple pendulum, k is expressed in terms of mass m, as
k ∝ m
\(\frac{m}{k}\) = Constant
Hence, the time period T, of a simple pendulum is independent of the mass of the bob.

(b) In the case of a simple pendulum, the restoring force acting on the bob of the pendulum is given as
F = -mg sinθ
where, F = Restoring force; m = Mass of the bob; g = Acceleration due to
gravity; θ = Angle of displacement
For small θ, sinθ ≈ θ
For large 0,sin0 is greater than 0.
This decreases the effective value of g.
Hence, the time period increases as
T = \(2 \pi \sqrt{\frac{l}{g}}\)
where, l is the length of the simple pendulum

(c) The time shown by the wristwatch of a man falling from the top of a tower is not affected by the fall. Since a wristwatch does not work on the principle of a simple pendulum, it is not affected by the acceleration due to gravity during free fall. Its working depends on spring action.

(d) When a simple pendulum mounted in a cabin falls freely under gravity, its acceleration is zero. Hence the frequency of oscillation of this simple pendulum is zero.

Question 17.
A simple pendulum of length l and having a bob of mass M is suspended in a car. The car is moving on a circular track of radius R with a uniform speed v. If the pendulum makes small f oscillations in a radial direction about its equilibrium position, what will be its time period?
Solution:
The bob of the simple pendulum will experience the acceleration due to gravity and the centripetal acceleration provided by the circular motion of the car.
Acceleration due to gravity = g
Centripetal acceleration = \(\frac{v^{2}}{R}\)
where, v is the uniform speed of the car R is the radius of the track
Effective acceleration (a_eff) is given as
a_eff = \(\sqrt{g^{2}+\left(\frac{v^{2}}{R}\right)^{2}}\)

Time period, T = \( 2 \pi \sqrt{\frac{l}{a_{e f f}}}\)
where, l is the length of the pendulum
∴ Time period, T = \(2 \pi \sqrt{\frac{l}{g^{2}+\frac{v^{4}}{R^{2}}}} \)

Question 18.
A cylindrical piece of cork of density of base area A and height h floats in a liquid of density ρ_l. The cork is depressed slightly and then released. Show that the cork oscillates up and down simple harmonically with a period
T = \(2 \pi \sqrt{\frac{\boldsymbol{h} \rho}{\rho_{\boldsymbol{l}} \boldsymbol{g}}}\)
where ρ is the density of cork. (Ignore damping due to viscosity of the liquid).
Solution:
Base area of the cork = A
Height of the cork = h
Density of the liquid = ρ_l
Density of the cork = ρ

In equilibrium:
Weight of the cork = Weight of the liquid displaced by the floating cork Let the cork be depressed slightly by x. As a result, some extra water of a certain volume is displaced. Hence, an extra up-thrust acts upward and provides the restoring force to the cork.
Up-thrust = Restoring force, F = Weight of the extra water displaced
F = -(Volume x Density x g)
Volume = Area x Distance through which the cork is depressed Volume = Ax
∴ F = -Ax ρ_lg ………………………… (i)
According to the force law,
F = kx
k = \(\frac{F}{x}\)

where k is a constant
k = \(\frac{F}{x}\) = -Aρ_lg ………………………………. (ii)
The time period of the oscillations of the cork,
T = \(2 \pi \sqrt{\frac{m}{k}} \) …………………………………… (iii)

where,
m = Mass of the cork
= Volume of the cork x Density
= Base area of the cork x Height of the cork x Density of the cork = Ahρ
Hence, the expression for the time period becomes
T = \(2 \pi \sqrt{\frac{A h \rho}{A \rho_{l} g}}\) = 2\(\pi \sqrt{\frac{h \rho}{\rho_{l} g}} \)

Question 19.
One end of a U-tube containing mercury is connected to a suction pump and the other end to atmosphere. A small pressure difference is maintained between the two columns. Show that, when the suction pump is removed, the column of mercury in the U-tube executes simple harmonic motion.
Solution:
Area of cross-section of the U-tube = A
Density of the mercury column = ρ
Acceleration due to gravity = g
Restoring force, F = Weight of the mercury column of a certain height
F = -(Volume x Density x g)
F = -(A x 2h x ρ x g) = -2Aρgh = -k x Displacement in one of the arms (h)

where, 2h is the height of the mercury column in the two arms
k is a constant, given by k = \(-\frac{F}{h}\) = 2Aρg
Time period = \(2 \pi \sqrt{\frac{m}{k}}=2 \pi \sqrt{\frac{m}{2 A \rho g}}\)
where, m is the mass of the mercury column
Let l be the length of the total mercury in the U-tube.
Mass of mercury, m = Volume of mercury x Density of mercury = Alρ
∴ T = \(2 \pi \sqrt{\frac{A l \rho}{2 A \rho g}}=2 \pi \sqrt{\frac{l}{2 g}} \)

Hence the mercury column executes simple harmonic motion with time period \(2 \pi \sqrt{\frac{l}{2 g}} \)

Additional Exercises

Question 20.
An air chamber of volume V has a neck area of cross-section a into which a ball of mass m just fits and can move up and down without any friction (see figure). Show that when the ball is pressed down a little and released, it executes SHM. Obtain an expression for the time period of oscillations assuming pressure-volume variations of air to be isothermal.

Solution:
Volume of the air chamber = V
Area of cross-section of the neck = a
Mass of the ball = m
The pressure inside the chamber is equal to the atmospheric pressure. Let the ball be depressed by x units. As a result of this depression, there would be a decrease in the volume and an increase in the pressure inside the chamber.
Decrease in the volume of the air chamber, ΔV = ax
Volumetric strain =
⇒ \(\frac{\Delta V}{V}=\frac{a x}{V}\)

Bulk Modulus of air, B = \(\frac{\text { Stress }}{\text { Strain }}=\frac{-p}{\frac{a x}{V}}\)
In this case, stress is the increase in pressure. The negative sign indicates that pressure increases with a decrease in volume.
p = \(\frac{-B a x}{V}\)
The restoring force acting on the ball,
F = p × a = \(\frac{-B a x}{V} \cdot a=\frac{-B a^{2} x}{V}\) ……………………………. (i)
In simple harmonic motion, the equation for restoring force is
F = -kx …………………………………….. (ii)
where, k is the spring constant Comparing equations (i) and (ii), we get
k = \(\frac{B a^{2}}{V}\)
Time period, T = \(2 \pi \sqrt{\frac{m}{k}}=2 \pi \sqrt{\frac{V m}{B a^{2}}}\)

Question 21.
You are riding in an automobile of mass 3000kg. Assuming that you are examining the oscillation characteristics of its suspension system. The suspension sags 15 cm when the entire automobile is placed on it. Also, the amplitude of oscillation decreases by 50% during one complete oscillation. Estimate the values of (a) the spring constant k and (6) the damping constant b for the spring and shock absorber system of one wheel, assuming that each wheel supports 750 kg.
Solution:
Mass of the automobile, m = 3000 kg
Displacement in the suspension system, x = 15cm = 0.15 m
There are 4 springs in parallel to the support of the mass of the automobile.
The equation for the restoring force for the system:
F = -4 kx = mg

where, k is the spring constant of the suspension system
Time period, T = \(2 \pi \sqrt{\frac{m}{4 k}}\)
and, k = \(\frac{m g}{4 x}=\frac{3000 \times 10}{4 \times 0.15}\) = 50000 = 5 x 10 ⁴N/m
Spring constant, k = 5 x 10⁴ N/m

Each wheel supports a mass, M = \(\frac{3000}{4}\) = 750 kg
For damping factor b, the equation for displacement is written as:
x = x₀e^-bt/2M

The amplitude of oscillation decreases by 50%

where, Time petiod,t =\(2 \pi \sqrt{\frac{m}{4 k}}=2 \pi \sqrt{\frac{3000}{4 \times 5 \times 10^{4}}}\) =0.7691s
∴ b =\(\frac{2 \times 750 \times 0.693}{0.7691}\) =1351.58kg/s
Therefore, the damping constant of the spring is 1351.58 kg/s.

Question 22.
Show that for a particle in linear SHM the average kinetic energy over a period of oscillation equals the average potential energy over the same period.
Solution:
The equation of displacement of a particle executing SHM at an instant t is given as
x = Asinωt
where,
A = Amplitude of oscillation
ω = Angular frequency = \(\sqrt{\frac{k}{M}}\)
The velocity of the particle is
ν = \(\frac{d x}{d t}\) = Aωcosωt

The kinetic energy of the particle is
E_k = \(\frac{1}{2} M v^{2}=\frac{1}{2} M A^{2} \omega^{2} \cos ^{2} \omega t \)
The potential energy of the particle is
E_p = \( \frac{1}{2} k x^{2}=\frac{1}{2} M \omega^{2} A^{2} \sin ^{2} \omega t\)

For time period T, the average kinetic energy over a single cycle is given as


And, average potential energy over one cycle is given as

It can be inferred from equations (i) and (ii) that the average kinetic energy for a given time period is equal to the average potential energy for the same time period.

Question 23.
A circular disc of mass 10 kg is suspended by a wire attached to its center. The wire is twisted by rotating the disc and released. The period of torsional oscillations is found to be 1.5s. The radius of the disc is 15 cm. Determine the torsional spring constant of the wire. (Torsional spring constant a is defined by the relation J = -αθ, where J is the restoring couple and 0 the angle of twist).
Solution:
Mass of the circular disc, m = 10 kg
Radius of the disc, r = 15cm = 0.15 m
The torsional oscillations of the disc has a time period, T = 1.5 s The moment of inertia of the disc is
I = \(\frac{1}{2}\) mr² = \(\frac{1}{2} \times(10) \times(0.15)^{2}\) = 0.1125kg-m²

Time period, T = \(2 \pi \sqrt{\frac{I}{\alpha}}\)
where, α is the torsional constant.
An21 4 x(3.14)2x 0.1125 , M
α = \(\frac{4 \pi^{2} I}{T^{2}}=\frac{4 \times(3.14)^{2} \times 0.1125}{(1.5)^{2}} \) = 1.972 N-m/rad
Hence, the torsional spring constant of the wire is 1.972 N-m rad^-1.

Question 24.
A body describes simple harmonic motion with amplitude of 5 cm and a period of 0.2 s. Find the acceleration and velocity of the body when the displacement is (a) 5 cm, (b) 3 cm, (c) 0 cm.
Solution:
Amplitude, A = 5 cm = 0.05m
Time period, T = 0.2 s
(a) For displacement, x = 5 cm = 0.05m
Acceleration is given by
a = -ω²x = \(-\left(\frac{2 \pi}{T}\right)^{2} x=-\left(\frac{2 \pi}{0.2}\right)^{2} \times 0.05\)
Velocity is given by
ν = ω \(\sqrt{A^{2}-x^{2}}=\frac{2 \pi}{T} \sqrt{(0.05)^{2}-(0.05)^{2}}\) = 0
When the displacement of the body is 5 cm, its acceleration is -5π² m/s² and velocity is 0.

(b) For displacement, x =3 cm = 0.03 m
Acceleration is given by
a = – ω²x = – \(\left(\frac{2 \pi}{T}\right)^{2}\)x = \(\left(\frac{2 \pi}{0.2}\right)^{2}\) 0.03 = -3π² m/s²
Velocity is given by

When the displacement of the body is 3 cm, its acceleration is -3π m/s² and velocity is 0.4π m/s.

(c) For displacement, x = 0
Acceleration is given by
a = – ω²x = 0
Velocity is given by

When the displacement of the body is 0, its acceleration is 0, and velocity is0.5π m/s.

Question 25.
A mass attached to a spring is free to oscillate, with angular velocity ω, in a horizontal plane without friction or damping. It is pulled to a distance x₀ and pushed towards the center with a velocity v₀ at time t = 0. Determine the amplitude of the resulting oscillations in terms of the parameters ω, x_0, and v₀. [Hint: Start with the equation x = a cos(ωt + θ) and note that the initial velocity is negative.]
Solution:
The displacement equation for an oscillating mass is given by
x = Acos(ωt + θ) …………………………… (i)
where A is the amplitude
x is the displacement
θ is the phase constant

Squaring and adding equations (iii) and (iv), we get

Hence, the amplitude of the resulting oscillation is \(\sqrt{x_{0}^{2}+\left(\frac{v_{0}}{\omega}\right)^{2}}\)