Q: When a block of mass M is suspended by a long wire of length L, the elastic potential energy stored in the wire is 21× stress × strain × volume. Show that it is equal to 21Mgl, where l is the extension. The loss in gravitational potential energy of the mass earth system is Mgl. Where does the remaining 21Mgl energy go?

Let the CSA of the wire be A. Stress = Area Force =AMg Strain =Ll Volume =AL We need to calculate the elastic potential energy stored in the wire which is given to be equal to 21× Stress × Strain × Volume. Elastic potential energy =21× Stress × Strain × Volume =21×AMg×Ll×AL=21Mgl The other 21Mgl 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: 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 = I Strain =Ll=1001⇒l=100L So, the increase in length will be 100L .Yes, the yield point is much less than the 1% strain because the human body consists of joints and not one uniform solid structure.

Question 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?

Accepted Answer

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.

Question 2

When a block of mass M is suspended by a long wire of length L, the elastic potential energy stored in the wire is 21​× stress × strain × volume. Show that it is equal to 21​Mgl, where l is the extension. The loss in gravitational potential energy of the mass earth system is Mgl. Where does the remaining 21​Mgl energy go?

Accepted Answer

Let the CSA of the wire be A.​ Stress = Area  Force ​=AMg​ Strain =Ll​ Volume =AL​ We need to calculate the elastic potential energy stored in the wire which is given to be equal to 21​×  Stress × Strain × Volume.​ Elastic potential energy =21​× Stress × Strain × Volume =21​×AMg​×Ll​×AL=21​Mgl​ The other 21​Mgl 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.

Question 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.

Accepted Answer

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.

Question 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?

Accepted Answer

​ Let my length =L Let the increase in length = I  Strain =Ll​=1001​⇒l=100L​​ So, the increase in length will be 100L​ .Yes, the yield point is much less than the 1% strain because the human body consists of joints and not one uniform solid structure.

Question 5

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

Accepted Answer

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

Question 6

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

Accepted Answer

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.

Question 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.

Accepted Answer

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.

Question 8

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

Accepted Answer

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.

Question 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?

Accepted Answer

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 byh=rρg2 Tcosθ​ Here, θ=90∘ ⇒h=rρg2 Tcos900​ ⇒h=0 Thus, the water level neither rises nor falls.

Question 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?

Accepted Answer

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 h=rρg2Tcosθ​ . 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  900 or 2700 .

Question 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.

Accepted Answer

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.

Question 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?

Accepted Answer

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.

Question 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?

Accepted Answer

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.

Question 14

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

Accepted Answer

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.

Question 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?

Accepted Answer

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.

Question 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?

Accepted Answer

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.

Question 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?

Accepted Answer

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.

Question 18

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

Accepted Answer

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 .

Question 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?

Accepted Answer

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 20

A rope 1 cm in diameter breaks if the tension in it exceeds 500 N. The maximum tension that may be given to a similar rope of diameter 2 cm is

Accepted Answer

F1​=500 N Let the required breaking force on the 2 cm wire be F . Breaking stress in 1 cm  wire = A1​F1​​=π(20.01​)2500​ Breaking stress in 2 cm wire = A2​F2​​=π(20.02​)2F2​​ The breaking stress is the same for a material.​⇒π(20.01​)2500​=π(20.02​)2F2​​⇒F2​=2000 N​

Chapter 14 : Some Mechanical Properties of Matter