Longitudinal Strain

Q.

What is linear elasticity?

Q. The length of a metal wire is $Z_1$, when the longitudinal strain in it is $T_1$ and when the longitudinal strain be $T_2$ then length of wire is $Z_2$ . The unstretched length of the wire is

1. $\sqrt{Z_1{Z}_2}$
   
2. $\frac{Z_1+Z_2}{2}$
   
3. $\frac{T_1{Z}_2-T_2{Z}_1}{T_1-T_2}$
   
4. $\frac{Z_1{T}_2+Z_2{T}_1}{T_2+T_1}$
   

Q. Each of the three blocks P, Q and R shown in figure have a mass of 3 kg. Each of the wires A and B has cross-sectional area 0.005 cm² and Young modulus 2 x 10¹¹ N m^-2. Neglect friction. Find the longitudinal strain developed in each of the wires. Take g = 10 m s^-2.

1. ${10}^{-4},{10}^{-4}$
2. $2\times {10}^{-4},4\times {10}^{-4}$
3. ${10}^{-4},2\times {10}^{-4}$
4. ${10}^{-4},4\times {10}^{-4}$

Q. A wire of cross-section area $2{\text{mm}}^2$ is stretched by $0.1\text{mm}$ by a certain force. How far (length) will the wire of same material and length but area $8{\text{mm}}^2$ stretch under the action of same force?

1. $0.25\text{mm}$
2. $0.005\text{mm}$
3. $0.025\text{mm}$
4. $0.05\text{mm}$

Q. A wire of initial length $L$ and radius $r$ is stretched by a length $l$. Another wire of same material but with initial length $2L$ and radius $2r$ is stretched by a length $2l$. The ratio of the stored elastic energy per unit volume in the first and second wire is

1. $1:4$
2. $1:2$
3. $2:1$
4. $1:1$

Q.

A 30kg hammer strikes a steel spike (nail) 2.30 cm in diameter while moving with speed 20.0 m/s. The hammer rebounds with speed 10.0 m/s after 0.110s. The average strain in the spike during the impact will be

($Y=2\times {10}^{11}N/m^2$)

1. ${10}^{-2}$

2. ${10}^{-3}$

3. ${10}^{-4}$

4. ${10}^{-5}$

Q. Two wires are made of the same material and have the same volume. However, wire $1$ has cross - sectional area $A$ and wire-$2$ has cross - sectional area $3A$. If the length of wire $1$ increases by $\Delta x$ on applying force $F$, how much force is needed to stretch wire $2$ by the same amount?

1. $F$
2. $4F$
3. $6F$
4. $9F$

Q.

A 30kg hammer strikes a steel spike (nail) 2.30 cm in diameter while moving with speed 20.0 m/s. The hammer rebounds with speed 10.0 m/s after 0.110s. The average strain in the spike during the impact will be

($Y=2\times {10}^{11}N/m^2$)

1. ${10}^{-2}$

2. ${10}^{-3}$

3. ${10}^{-4}$

4. ${10}^{-5}$

Q. Two wires of same material and radius have their lengths in ratio 1 : 2. If these wires are stretched by the same force, the strain produced in the two wires will be in the ratio

1. 2 : 1
2. 1 : 1
3. 1 : 2
4. 1 : 4

Q. A wire of cross section $A$ is stretched horizontally between two clamps located $2lm$ apart. A weight $Wkg$ is suspended from the mid-point of the wire. If the mid-point sags vertically through a distance $x<<l$, the strain produced is

1. $\frac{2x^2}{l^2}$
2. $\frac{x^2}{l^2}$
3. $\frac{x^2}{2l^2}$
4. None of these

Q. Each of the pictures shows four objects tied together with rubber bands being pulled to the right across a horizontal frictionless surface by a force $F$. All the objects have the same mass. All the rubber bands obey Hooke’s law and have the same equilibrium length and the same force constant. Which of the pictures is drawn most correctly?

1. 
2. 
3. 
4. 

Q.

Two wires A and B of the same material have their lengths in the ratio of 1:2 and their diameters in the ratio of 2:1. If they are stretched with the same force, the ratio of the increase in the length of A to that of B will be

1. 1:2

2. 4:1

3. 1:8

4. 1:4

Q. In an experiment, brass and steel wires of length $1\text{m}$ each with areas of cross-section $1{\text{mm}}^2$ are used. The wires are connected in series and one end of the combined wire is connected to a rigid support and other end is subjected to elongation. The stress requires to produce a net elongation of $0.2\text{mm}$ is
[Take Young’s modulus for steel and brass as $120\times {10}^9{\text{N/m}}^2$ and $60\times {10}^9{\text{N/m}}^2$respectively]

1. $1.2\times {10}^6{\text{N/m}}^2$
2. $0.2\times {10}^6{\text{N/m}}^2$
3. $8\times {10}^6{\text{N/m}}^2$
4. $4.0\times {10}^6{\text{N/m}}^2$

Q.

For a fixed amount, when the rate of interest is doubled and the time becomes 5 times, the interest becomes

1. $10times$

2. double

3. treble

4. $4times$

Q. Two wires A and B are made of same material. The wire A has a length l and diameter r while the wire B has length 2l and diameter 2r. If the two wires are stretched by the same force, the elongation in A divided by elongation in B is

1. 1/8

2. 1/4

3. 4

4. 8

Q. A wire increases by ${10}^{-3}$ times of its length when a stress of $1\times {10}^8{\text{Nm}}^{-2}$ is applied to it in the longitudinal direction. Ratio of longitudinal stress and longitudinal strain in the wire is

1. ${10}^{11}{\text{Nm}}^{-2}$
2. ${10}^8{\text{Nm}}^{-2}$
3. $2\times {10}^6{\text{Nm}}^{-2}$
4. $5.2\times {10}^4{\text{Nm}}^{-2}$

Q. Each of the three blocks P, Q and R shown in figure have a mass of 3 kg. Each of the wires A and B has cross-sectional area 0.005 cm² and Young modulus 2 x 10¹¹ N m^-2. Neglect friction. Find the longitudinal strain developed in each of the wires. Take g = 10 m s^-2.

1. ${10}^{-4},{10}^{-4}$
2. $2\times {10}^{-4},4\times {10}^{-4}$
3. ${10}^{-4},2\times {10}^{-4}$
4. ${10}^{-4},4\times {10}^{-4}$

Q. A wire is stretched by 0.01 m by a certain force F. Another wire of same material whose diameter and length are double to the original wire is stretched by the same force. Then its elongation will be

1. 0.005 m
2. 0.01 m
3. 0.02 m
4. 0.002 m

Q. If the ratio of lengths, radii and Young's modulus of steel and brass wires in the figure are $a,b$ and $c$ respectively, then the corresponding ratio of increase in their lengths would be

1. $\frac{2a^{2}c}{b}$
2. $\frac{3a}{2b^{2}c}$
3. $\frac{2ac}{b^2}$
4. $\frac{3c}{2a{b}^2}$

Q. A force of $5\times {10}^3\text{N}$ is applied on the rod of length $1\text{m}$. Find the linear strain if elongation in the rod is $10\text{mm}$.

1. $0.1$
2. $1$
3. $0.01$
4. $0.001$

Q. Find the displacement ( in $\text{mm}$) of the midpoint of a steel rod of length $2\text{m}$ and mass $10\text{kg}$ due to its own weight

[Assume, modulus of elasticity $Y=2\times {10}^{11}{\text{N/m}}^2$, area of cross-section $A={10}^{-8}{\text{m}}^2$]

Q. A 6-kg weight is fastened to the end of a steel wire of un-stretched length 60 cm. It is whirled in a vertical circle and has an angular velocity of 2 revolutions per second at the bottom of the circle. The area of cross-section of the wire is $0.05c{m}^2$. Calculate the elongation of the wire when the weight is at the lowest point of the path. Young's modulus of steel = $2\times {10}^{11}$ Pa.

1. 4.97 $\times$${10}^{-5}$m

2. 3.77 $\times$${10}^{-4}$m

3. 2.34$\times$${10}^{-4}$m

4. 2.34$\times$${10}^{-5}$m

Q. An Indian rubber cord $L\text{m}$ long and area of cross-section $A{\text{m}}^2$ is suspended vertically. Density of rubber is $D{\text{kg/m}}^3$ and Young's modulus of rubber is $E{\text{N/m}}^3.$ If the wire extends by $l\text{m}$ under its own weight, then extension $l$ is

1. $L^{2}Dg/E$
2. $L^{2}Dg/2E$
3. $L^{2}Dg/4E$
4. $L$

Q. The graph shows the behavior of the length of a wire in the region for which the material obeys Hooke’s law, then $P$ and $Q$ represent

1. P = applied force, Q = extension
2. P = extension, Q = applied force
3. P = extension, Q = stored elastic energy
4. P = stored elastic energy, Q = extension

Q. A wooden wheel of radius $R$ is made of two semicircular parts (see figure). The two parts are held together by a ring made of a metal strip of cross - sectional area $S$ and length $L$. $L$ is slightly less than $2\pi R$. To fit the ring on the wheel, it is heated so that its temperature rises by $\Delta T$ and it just steps over the wheel. As it cools down to surrounding temperature, it presses the semicircular parts together. If the coefficient of linear expansion of the metal is $\alpha$ and its Young's modulus is $Y$, the force that one part of the wheel applies on the other part is-

1. $2\pi SY\alpha \Delta T$
2. $SY\alpha \Delta T$
3. $\pi SY\alpha \Delta T$
4. $2SY\alpha \Delta T$

Q. Two identical wires of rubber and iron are stretched by the same weight, then the number of atoms in the iron wire will be

1. equal to that of rubber
2. less than that of the rubber
3. more than that of rubber
4. none of the above

Q. Three blocks, each of same mass m, are connected with wires $W_1$ and $W_2$ of same cross- sectional area a and Young's modulus Y. Neglecting friction the strain developed in wire $W_2$ is

1. $\frac{2}{3}\frac{mg}{aY}$
2. $\frac{3}{2}\frac{mg}{aY}$
3. $\frac{1}{3}\frac{mg}{aY}$
4. $\frac{3mg}{aY}$

Q. Two persons pull a rope towards themselves. Each person exerts a force of $100N$ on the rope. The Young’s modulus of the material of the rope, if it extends in length by $1cm$ is ${10}^n$. Then value of $n$ is

Original length of the rope $=2m$ and the area of cross-section $2{cm}^2$.

Q. The Young’s modulus of steel is twice as that of brass. Two wires of same length and of same area of cross section, one of steel and another of brass are suspended from the same roof. If we want the lower ends of the wires to be at the same level, then the weights added to the steel and brass wires must be in the ratio of

1. $2:1$
2. $4:1$
3. $1:1$
4. $1:2$

Q. A 6-kg weight is fastened to the end of a steel wire of un-stretched length 60 cm. It is whirled in a vertical circle and has an angular velocity of 2 revolutions per second at the bottom of the circle. The area of cross-section of the wire is $0.05c{m}^2$. Calculate the elongation of the wire when the weight is at the lowest point of the path. Young's modulus of steel = $2\times {10}^{11}$ Pa.

1. 4.97 $\times$${10}^{-5}$m

2. 3.77 $\times$${10}^{-4}$m

3. 2.34$\times$${10}^{-4}$m

4. 2.34$\times$${10}^{-5}$m