## WBCHSE Class 11 Physics On Elasticity Short Questions And Answers

**Question 1. A spring is cut into two equal pieces. What is the spring the constant of each part if the spring constant of the original spring is k,**

**Solution:**

Let us consider that the spring elongates by x when a force F is applied on it. So, the force constant of the spring, k = F/x.

Now, if the spring is cut into two equal parts, then on the application of the same force F, each part of the spring will elongate by x/2.

The force constant each part, \(k^{\prime}=\frac{F}{\frac{x}{2}}=\frac{2 F}{x}=2 k\)

**Question 2. A spring having spring constant k is cut into two parts in the ratio 1:2. Find the spring constants of the two parts.**

**Solution:**

Let the initial length of the spring be x.

The spring constant of a particular spring is inversely proportional to its length.

∴ kx = constant.

When the spring is cut into two parts in the ratio 1:2, the length of the two parts are x/3 and 2x/3 respectively.

⇒\(k_1 \frac{x}{3}=k x \text { or, } k_1=3 k\)

and \(k_2 \cdot \frac{2 x}{3}=k x \text { or, } k_2=\frac{3 k}{2}\)

**Question 3. The length of a metal wire is L _{1}. when the tension is T_{1} and L_{2} when the tension is T_{2} The unstretched length of the wire is**

- \(\frac{L_1+L_2}{2}\)
- \(\sqrt{L_1 L_2}\)
- \(\frac{T_2 L_1-T_1 L_2}{T_2-T_1}\)
- \(\frac{T_2 L_1+T_1 L_2}{T_2+T_1}\)

**Answer:**

Young’s modulus, Y = \(\frac{\text { stress }}{\text { strain }}\)

or, strain = \(\frac{\text { stress }}{\text { strain }}\)

If the length of the wire is L_{0} when there is no tension in the string, then in the first case, stress = \(\frac{T_1}{\alpha}\) and strain = \(\frac{L_1-L_0}{L_0}\)

[a = area of cross-section = constant (approximately)]

So, \(\frac{L_1-L_0}{L_0}=\frac{T_1}{\alpha Y} \quad \text { or, } \frac{1}{\alpha Y}=\frac{1}{T_1}\left(\frac{L_1}{L_0}-1\right)\)

Similarly in the second case, \(\frac{1}{\alpha Y}=\frac{1}{T_2}\left(\frac{L_2}{L_0}-1\right)\)

So, \(\frac{1}{T_1}\left(\frac{L_1}{L_0}-1\right)=\frac{1}{T_2}\left(\frac{L_2}{L_0}-1\right)\)

or, \(\frac{1}{L_0}\left(\frac{L_1}{T_1}-\frac{L_2}{T_2}\right)=\frac{1}{T_1}-\frac{1}{T_2}\)

or, \(\frac{1}{L_0} \frac{T_2 L_1-T_1 L_2}{T_1 T_2}=\frac{T_2-T_1}{T_1 T_2}\)

∴ \(L_0=\frac{T_2 L_1-T_1 L_2}{T_2-T_1}\)

The option 3 is correct

**Question 4. A liquid of bulk modulus k is compressed by applying an external pressure such that its density increases by 0.04%. The pressure applied to the liquid is**

- k/10000
- k/10000
- 1000k
- 0.01k

**Answer:**

k = \(\frac{p}{\frac{\Delta V}{V}}\)

or, \(p=k \times \frac{\Delta V}{V}=k \times \frac{\Delta \rho}{\rho}=k \times 0.01 \%=\frac{k}{10000}\)

The option 1 is correct.

**Question 5. The stress along the length of a rod (with a rectangular cross section) is 1% of the Young’s modulus of its material. What is the approximate percentage of change in its volume? (Poisson’s ratio of the material of the rod is 0.3)**

- 3%
- 1%
- 0.7%
- 0.4%

**Answer:**

Let, the volume of the rod, V = xyz, and Young’s modulus of its material of the rod = Y

Now, \(\frac{F}{A}=Y \times 1 \%\)

or, \(Y \times \frac{\Delta x}{x}=\frac{Y}{100}\)

or, \(\frac{\Delta x}{x}=0.01\)

∴ \(\frac{\Delta V}{V}=\frac{\Delta x}{x}+\frac{\Delta y}{y}+\frac{\Delta z}{z}\)

= \(\frac{\Delta x}{x}-\sigma \frac{\Delta x}{x}-\sigma \frac{\Delta x}{x}\) ……..(1)

[Poisson’s ratio, \(\sigma=\frac{\text { lateral strain }}{\text { longitudinal strain }}=\frac{\frac{\Delta y}{y}}{\frac{\Delta x}{x}}=\frac{\frac{\Delta z}{z}}{\frac{\Delta x}{x}}\)]

The negative symbol in equation (1) implies that, as length increases due to stress, the value of y and z decreases simultaneously.

∴ From equation (1),

∴ \(\frac{\Delta V}{V}=0.01-2 \times 0.3 \times 0.01=0.004=0.4 \%\)

The option 4 is correct.

**Question 6. When a rubber band is stretched by a distance x, it exerts a restoring force of magnitude F = ax + bx ^{2}, where a and b are constants. The work done in stretching the unstretched rubber band by L isothermal**

- \(a L^2+b L^3\)
- \(\frac{1}{2}\left(a L^2+b L^3\right)\)
- \(\frac{a L^2}{2}+\frac{b L^3}{3}\)
- \(\frac{1}{2}\left(\frac{a L^2}{2}+\frac{b L^3}{3}\right)\)

**Answer:**

⇒ \(\int d W=\int F d l\)

W = \(\int_0^L a x d x+\int_0^L b x^2 d x=\frac{a L^2}{2}+\frac{b L^3}{3}\)

The option 3 is correct.

**Question 7. A man grows into a giant such that his linear dimensions increase by a factor of 9. Assuming that his density remains the same, the stress in the leg will change by a factor of**

- 9
- 1/9
- 81
- 1/81

**Answer**:

According to the question, \(\frac{V_f}{V_i}=(9)^3\)

So, \(\frac{m_f}{m_i}=(9)^3\)

Also, \(\frac{A_f}{A_i}=(9)^2\)

Stress = \(\frac{\text { force }}{\text { area }}=\frac{m \times g}{A}\)

∴ \(\frac{S_f}{S_i}=\frac{m_f}{m_i} \times \frac{A_i}{A_f}=(9)^3 \times \frac{1}{(9)^2}=9\)

The option 1 is correct.

**Question 8. An external pressure P is applied on a cube at 0°C so that it is equally compressed from all sides. K is the bulk modulus of the material of the cube and α is its coefficient of linear expansion. Suppose we want to bring the cube to its original size by heating it. The temperature should be raised by**

- \(\frac{P}{3 \alpha K}\)
- \(\frac{P}{\alpha K}\)
- \(\frac{3 \alpha}{P K}\)
- \(3 P K \alpha\)

**Answer:**

Bulk modulus, K= \(\frac{P}{\left(\frac{\Delta V}{V}\right)}\)

∴ \(\frac{\Delta V}{V}=\frac{P}{K}[\Delta V= change in volume]\)

If we bring back the cube to its original size by increasing the temperature Δt,

⇒ \(\Delta V=V \cdot \gamma \Delta t\)

or, \(\Delta t=\frac{\Delta V}{V} \cdot \frac{1}{\gamma}=\frac{\Delta V}{V} \cdot \frac{1}{3 \alpha}=\frac{P}{3 k \alpha}\)

The option (1) is correct.

**Question 9. A solid sphere of radius r made of a soft material of bulk modulus K is surrounded by a liquid in a cylindrical container. A massless piston of area floats on the surface of the liquid, covering the entire cross-section of the cylindrical container. When a mass m is placed on the surface of the piston to compress the liquid, the fractional decrement in the radius of the sphere, (dr/r) is**

- \(\frac{m g}{3 K a}\)
- \(\frac{m g}{K a}\)
- \(\frac{K a}{m g}\)
- \(\frac{K a}{3 m g}\)

**Answer:**

Bulk modulus, K = \(-V \frac{d p}{d V}\)

or, \(-\frac{d V}{V}=\frac{d p}{K}\)

or, \(\frac{-3 d r}{r}=\frac{\frac{m g}{a}}{K}\left[because V=\frac{4}{3} \pi r^3\right]\)

or, \(\frac{d r}{r}=-\frac{m g}{3 K a} \quad therefore\left|\frac{d r}{r}\right|=\frac{m g}{3 K a}\)

The option 1 is correct

**Question 10. The copper of fixed volume V is drawn into a wire of length l. When this wire is subjected to a constant force F, the extension produced in the wire is Δl. Which of the following graphs is a straight line?**

- Δl versus 1/l
- Δl versus l
^{2} - Δl versus 1/l
^{2} - Δl versus l

**Answer:**

Y = \(\frac{F l}{A \Delta l} \text { or, } \Delta l=\frac{F l}{A Y}=\frac{F l^2}{V Y}\)

∴ \(\Delta l \propto l^2\)

The option 2 is correct.

**Question 11. The approximate depth of an ocean is 2700 m. The compressibility of water is 45.4 x 10 ^{-11}^{ }Pal^{-1} and the density of water is 10^{3}kg/m^{3}. What fractional compression of water will be obtained at the bottom of the ocean?**

- 0.8 x 10
^{-2} - 1.0 x 10
^{-2} - 1.2 x 10
^{-2} - 1.4×10
^{-2}

**Answer:**

Due to AP amount of increase in pressure, there is AV

amount of compression in volume V.

So, fractional compression = \(\frac{\Delta V}{V}\)

and compressibility, K = \(\frac{1}{V} \frac{\Delta V}{\Delta P}\)

Now consider the magnitude, \(\frac{\Delta V}{V}=K \Delta P\)

Here, ΔP = hρg = 2700 x 10^{3} x 10 Pa [taking g = 10m/s^{2}]

Hence, fractional compression, \(\frac{\Delta V}{V} =\left(45.4 \times 10^{-11}\right) \times\left(2700 \times 10^3 \times 10\right)\)

= \(1.226 \times 10^{-2}\)

The option 3 is correct.

**Question 12. The density of a metal at normal pressure is p. Its density when it is subjected to an excess pressure p is p’. If B is the bulk modulus of the metal, the ratio of \(\frac{e^{\prime}}{\rho}\)**

- \(1+\frac{B}{p}\)
- \(\frac{1}{1-\frac{p}{B}}\)
- \(1+\frac{p}{B}\)
- \(\frac{1}{1+\frac{P}{B}}\)

**Answer:**

Volume strain = change in pressure = p

Initial volume, V = \(\frac{M}{\rho}\)

Final volume, \(V^{\prime}=\frac{M}{\rho^{\prime}}\)

Change in volume, \(V^{\prime}-V=M\left(\frac{\rho-\rho^{\prime}}{\rho^{\prime} \rho}\right)\)

∴ Volume strain = \(=\frac{V^{\prime}-V}{V}=\frac{\rho-\rho^{\prime}}{\rho^{\prime}}\)

∴ \(B=-\frac{p V}{V^{\prime}-V}=-\frac{p \times \rho^{\prime}}{\rho-\rho^{\prime}}\)

or, \(\underset{B}{p}=-\frac{\rho-\rho^{\prime}}{\rho^{\prime}}=\frac{\rho^{\prime}-\rho}{\rho^{\prime}}\)

or, \(\frac{\rho}{\rho^{\prime}}=1-\frac{p}{B}\)

∴ \(\frac{\rho^{\prime}}{\rho}=\frac{1}{1-\frac{p}{B}}\)

The option 2 is correct.

**Question 12. Two wires are made of the same material and have the same volume. The first wire has cross-sectional area A and the second wire has cross-sectional area 3A. If the length of the first wire is increased by Δl. on applying a force F, how much force is needed to stretch the second wire by the same amount?**

- 4F
- 6F
- 9F
- F

**Answer:**

In case of first wire, Y = \(\frac{F / A}{\Delta l / l_0}=\frac{F l_0}{A \Delta l}\)

or, \(F=\frac{Y A \Delta l}{l_0}\)

In the case of the second wire,

Y = \(\frac{F^{\prime} / 3 A}{\frac{\Delta l}{l_0 / 3}}=\frac{F^{\prime} l_0}{9 A \Delta l}\)

or, \(F^{\prime}=\frac{9 Y A \Delta l}{l_0}=9 F\)

The option 3 is correct.

**Question 13. Which type of substances are called elastomers? Give one example.**

**Answer:**

Elastomers are those materials for which stress-strain variation is not a straight line within the elastic limit. An elastomer is a polymer with viscoelasticity (colloquially elasticity), generally having low Young’s modulus and high failure strain compared with other materials.

**Example:** Rubber.

**Question 14. Bridges are declared unsafe after long use. Why?**

**Answer:**

A bridge undergoes alternating stress and strain a large number of times during its use. A bridge loses its elastic strength when it is used for a long time. Therefore, the amount of strain for a given stress will become large and ultimately the bridge will collapse. So, they are declared unsafe after long use.

**Question 15. What are elastomers? Give two examples for the same.**

**Answer:**

Elastomers (elastic polymers) are materials of low Young’s modulus but of very high elastic limits. Such a material can withstand high strain but can still develop sufficient stress to bring it back to its initial size and shape.

**Examples:** natural rubber, thermoplastics.

**Question 16. What is the value of rigidity modulus of elasticity for an incompressible liquid?**

**Answer:**

A liquid, compressible or incompressible, does not have any defined shape; it cannot withstand shear. So it can never generate any shearing stress. Hence the rigidity modulus of elasticity of a liquid is zero.

**Question 17. Which type of energy is stored in the spring of wrist wristwatch?**

**Answer:**

Potential energy is stored in the spring of wrist watch.

**Question 18. The stress-strain graph for materials A and B are as shown in the graphs drawn to the same scale, which graph represents a property of ductile materials? Justify your answer.**

**Answer:**

The graph for material A represents the property of ductile material because of its greater plastic range.

**Question 19. Two wires A and B of length l, radius r, and length 21, radius 2 r having the same Young’s modulus Y are hung with a weight of mg as shown in the figure. What is the net elongation in the two wires?**

**Answer:**

The length and radius of wire A are l and r and that of wire B are 2l and 2 r respectively.

If l_{1} and l_{2} are the individual elonga¬tion of wire A and wire B, then the net elongation,

∴ \(\Delta l =\Delta l_1+\Delta l_2=\frac{m g l}{\pi r^2 Y}+\frac{m g(2 l)}{\pi(2 r)^2 Y}\)

= \(\left(\frac{m g l}{\pi r^2 Y}+\frac{2 m g l}{4 \pi r^2 Y}\right)=\frac{4 m g l+2 m g l}{4 \pi r^2 Y}=\frac{3}{2} \frac{m g l}{\pi r^2 Y}\)

**Question 20. Which of the two forces-deforming or restoring is responsible for the elastic behavior of a substance?**

**Answer: **Restoring force is responsible for the elastic behavior of a substance.