HC Verma Questions and Solutions: Chapter 14: Some Mechanical Properties of Matter- 1 | HC Verma Solutions - JEE PDF Download

Q.1. The ratio stress/strain remain constant for small deformation of a metal wire. When the deformation is made larger, will this ratio increase or decrease?

The ratio of stress to strain will decrease.
Beyond the elastic limit, the body loses its ability to restore completely when subjected to stress. Thus, there occurs more strain for a given stress. At some points, however, the body undergoes strain without the application of stress. So, the ratio of stress to strain decreases. 

Q.2. When a block a mass M is suspended by a long wire of length L, the elastic potential potential energy stored in the wire is (1/2) × stress × strain × volume. Show that it is equal to (1/2) Mgl, where l is the extension. The loss in gravitational potential energy of the mass earth system is Mgl. Where does the remaining (1/2) Mgl energy go? 

Let the CSA of the wire be A.

 Strain = I/L
Volume = AL
We need to calculate the elastic potential energy stored in the wire which is given to be equal to (1/2) x Stress x Strain x  Volume.
Elastic potential energy = (1/2) x Stress x Strain x  Volume


The other   is converted into kinetic energy of the mass.
When the mass leaves its initial point on the spring, it acquires a velocity as it moves down. The velocity reaches its maximum at the end point. The spring oscillates. Finally, when the kinetic energy is dissipated into heat, the spring comes to rest.

Q.3. When the skeleton of an elephant and the skeleton of a mouse are prepared in the same size, the bones of the elephant are shown thicker than those of the mouse. Explain why the bones of an elephant are thicker than proportionate. The bones are expected to withstand the stress due to the weight of the animal.

The elephant has a greater weight than a mouse, but the material that makes their bones is the same. This means that in order to sustain an elephant's weight, one's bones need to suffer less stress. Stress = Force/area. A greater cross-sectional area reduces stress on the bones. This is why an elephant's bones are thicker.  

Q.4. The yield point of a typical solid is about 1%. Suppose you are lying horizontally and two persons are pulling your hands and two persons are pulling your legs along your own length. How much will be the increase in your length if the strain is 1% ? Do you think your yield point is 1% or much less than that?

Let my length = L
Let the increase in length = l
Strain

So, the increase in length will be L/100.
Yes, the yield point is much less than the 1% strain because the human body consists of joints and not one uniform solid structure.

Q.5. When rubber sheets are used in a shock absorber, what happens to the energy of vibration?

The energy of vibration dissipates as heat from the shock absorber.

Q.6. If a compressed spring is dissolved in acid, what happened to the elastic potential energy of the spring?

When a compressed spring dissolves in an acid, the acid molecules leave the sold lattice of the spring faster than the uncompressed spring. This in turn increases the kinetic energy of the solution. As a result, the temperature of the acid also increases. However, this temperature increase will be very small because the mechanical energy content in the spring is lesser than its chemical energy content.

Q.7. A steel blade placed gently on the surface of water floats on it. If the same blade is kept well inside the water, it sinks. Explain.

 It floats because of the surface tension of water. The surface of water behaves like a stretched membrane. When a blade is placed on the water surface, it's unable to pierce the stretched membrane of water due to its low weight and remains floating. 

However, if the blade is placed below the surface of water, it no longer experiences the surface tension and sinks to the bottom as the density of the blade is greater than that of water. 

Q.8. When some wax is rubbed on a cloth, it becomes waterproof. Explain.

A liquid wets a surface when the angle of contact of the liquid with the surface is small or zero. Due to its fibrous nature, cloth produces capillary action when in contact with water. This makes clothes have very small contact angles with water. When wax is rubbed over cloth, the water does not wet the cloth because wax has a high contact angle with water.

Q.9. The contact angle between pure water and pure silver is 90°. If a capillary tube made of silver is dipped at one end in pure water, will the water rise in the capillary?

No, the water will neither rise nor fall in the silver capillary.

According to Jurin's law, the level of water inside a capillary tube is given by

Thus, the water level neither rises nor falls.

Q.10. It is said that a liquid rises or is depressed in capillary due to the surface tension. If a liquid neither rises nor depresses in a capillary, can we conclude that the surface tension of the liquid is zero?

No, we cannot conclude the surface tension to be zero solely by the fact that the liquid neither rises nor falls in a capillary.
The height of the liquid inside a capillary tube is given by   From the equation, we see that the height (h) of the liquid may also be zero if the contact angle θ between the liquid and the capillary tube is 90° or 270°.

Q.11. The contact angle between water and glass is 0°. When water is poured in a glass to the maximum of its capacity, the water surface is convex upward. The angle of contact in such a situation is more than 90°. Explain.

When water is poured in a glass, it reaches the brim and rises further. The edge of the glass lies below the water level. In this case, the force of attraction due to molecules of the glass surface is not perpendicular to the solid. Here, the contact angle can be greater than the standard contact angle for a pair of substances.

Q.12. A uniform vertical tube of circular cross section contains a liquid. The contact angle is 90°. Consider a diameter of the tube lying in the surface of the liquid. The surface to the right of this diameter pulls the surface on the left of it. What keeps the surface on the left in equilibrium?

 As the angle of contact is 0, there is no force between the surface of the tube and the liquid. The diameter of the liquid surface is pulled on both sides by equal and opposite forces of surface tension. This results in no net force remaining on the surface of the liquid. Hence, the liquid stays in equilibrium. 

Q.13. When a glass capillary tube is dipped at one end in water, water rises in the tube. The gravitational potential energy is thus increased. Is it a violation of conservation of energy?

No, it does not violate the principle of conservation of energy.
There is a force of attraction between glass and water, which is why the liquid rises in the tube. However, when water and glass are not in contact, there exists a potential energy in the system. When they are brought into contact, this potential energy is first converted into kinetic energy, which lets the liquid rush upwards in the tube, and then into gravitational potential energy. Therefore,  energy is not created in the process.

Q.14. If a mosquito is dipped into water and released, it is not able to fly till it is dry again. Explain?

A mosquito thrown into water has its wings wetted. Now, wet wing surfaces tend to stick together because of the surface tension of water. This does not let the mosquito fly.

Q.15. The force of surface tension acts tangentially to the surface whereas the force due to air pressure acts perpendicularly on the surface. How is then the force due to excess pressure inside a bubble balanced by the force due to the surface tension?

The forces act tangentially to the bubble surface on both sides of a given line but they have one component normal to the bubble surface. This component balances the force due to excess pressure inside the bubble.
In the figure, let us consider a small length AB on the surface of the spherical bubble. Let the surface forces act tangentially along A and B. On producing the forces backwards, they meet at a point O. By the parallelogram law of forces, we see that the resultant force acts opposite to the normal. This balances the internal forces due to excess pressure.

Q.16. When the size of a soap bubble is increased by pushing more air in it, the surface area increases. Does it mean that the average separation between the surface molecules is increased?

No. The average intermolecular distances do not increase with an increase in the surface area.
A soap bubble's layer consists of several thousand layers of molecules. An increase in the surface area causes the surface energy to also increase. This in turn allows more and more molecules from the inner liquid layers of the bubble to attain potential energy, enabling them to enter the outer surface of the bubble. Hence, the surface area increases.

Q.17. Frictional force between solids operates even when they do not move with respect to each other. Do we have viscous force acting between two layers even if there is no relative motion?

No. For a liquid at rest, no viscous forces exist.
Viscous forces oppose relative motion between the layers of a liquid. These layers do not exist in a liquid that is at rest. Therefore, it is obvious that viscous forces are non-existent in a static liquid. 

Q.18. Water near the bed of a deep river is quiet while that near the surface flows. Give reasons.

The motion of any liquid is dependent upon the amount of stress acting on it. The motion of one layer of liquid is resisted by the other due to the property of viscosity. A river bed remains in a static state. Therefore, any immediate layer of liquid in contact with the river bed will also remain static due to the frictional force. However, the next layer of liquid above this static layer will have a greater velocity due to lesser resistance offered by the static layer. Moving upwards, subsequent layers provide lesser and lesser resistance to the movement of the layers above it. Finally, the topmost layer acquires the maximum velocity. Therefore, for a river, the surface waters flow the fastest.

Q.19. If water in one flask and castor oil in other are violently shaken and kept on a table, which will come to rest earlier?

Castor oil will come to rest more quickly because it has a greater coefficient of viscosity than water. Castor oil has a higher viscosity than water. It will therefore, lose kinetic energy and come to rest faster than water. 

Question for HC Verma Questions and Solutions: Chapter 14: Some Mechanical Properties of Matter- 1

Try yourself:Figure shows a capillary tube of radius r dipped into water. If the atmospheric pressure is P₀, the pressure at point A is

• A.

  P₀
• B.

  
• C.

  
• D.

  

Explanation

Here:
Radius of the tube = r
Net upward force due to surface tension  = S  cosθ x 2πr
Upward pressure 
Net downward pressure due to atmosphere = P₀
⇒ Net pressure at A = 
Since θ is small,
cosθ ≈ 1.
⇒ Net pressure = 

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Question for HC Verma Questions and Solutions: Chapter 14: Some Mechanical Properties of Matter- 1

Try yourself:The viscous force acting between two layers of a liquid is given by   This F/A may be called 

• A.

  pressure
• B.

  longitudinal stress
• C.

  tangential stress
• D.

  volume stress

Explanation

The viscous force acts tangentially between two parallel layers of a liquid. In terms of force on a material, it is analogous to a shearing force. 

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Question for HC Verma Questions and Solutions: Chapter 14: Some Mechanical Properties of Matter- 1

Try yourself:A spherical ball is dropped in a long column of a viscous liquid. The speed of the ball as a function of time may be best represented by the graph 

• A.

  A
• B.

  B
• C.

  C
• D.

  D

Explanation

Initially, when the ball starts moving, its velocity is small. Gradually, the velocity of the ball increases due to acceleration caused by gravity. However, as the velocity increases, the viscous force acting on the ball also increases. This force tends to decelerate the ball. Therefore, after reaching a certain maximum velocity, the ball slows down. 

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*Multiple options can be correct

Question for HC Verma Questions and Solutions: Chapter 14: Some Mechanical Properties of Matter- 1

Try yourself:A student plots a graph from his reading on the determination of Young modulus of a metal wire but forgets to put the labels. the quantities on X and Y-axes may be respectively

• A.

  weight hung and length increased
• B.

  stress applied and length increased
• C.

  stress applied and strain developed
• D.

  length increased and the weight hung.

Explanation

All options are correct. 
(a) When a weight is loaded on a wire, the length of the wire increases. The relationship between weight and length is linear.
(b) When a weight is loaded, it produces stress on the wire. The relationship between stress and increase in length is also linear.
(c) When stress is applied, strain develops. Therefore, both are linearly related. 
(d) Since the value of Y  for the wire is unknown, X may also be the increase in its length. Nevertheless, they still show the same linear relationship.  

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Question for HC Verma Questions and Solutions: Chapter 14: Some Mechanical Properties of Matter- 1

Try yourself:A liquid is contained in a vertical tube of semicircular cross section. The contact angle is zero. The force of surface tension on the curved part and on the flat part are in ratio

• A.

  1:1
• B.

  1:2
• C.

  π:2
• D.

  2:π

Explanation

Let the height of the liquid-filled column be L.
Let the radius be denoted by R.
Total perimeter of the curved part = semi - circumference of upper area = πr
Total surface tension force = πRS
Total perimeter of the flat part = 2R
Total surface tension force = 2RS
Ratio of curved surface force to flat surface force = 

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