Elastic Potential Energy

Q. A block is attached to a spring having equilibrium position at $x=0$. The block oscillates between $x=-4\text{m}$ and $x=4\text{m}$. Find the change in potential energy stored in the spring when the block moves from $x=2$ to $x=-3$. Given the spring constant is $5\text{N/m}$.

1. $12.5\text{J}$
2. $32.5\text{J}$
3. $37.5\text{J}$
4. $10\text{J}$

Q. A body of mass $m$ hangs by as inextensible light string that passes over a smooth massless pulley that is fitted with a light spring of stiffness $k$ as shown in the figure. If the body is released from rest, calculate the maximum elongation of the spring. Take $g=10{\text{m/s}}^2$.

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

Q. A spring of natural length ${}^{\prime }{x}^{\prime }\text{m}$ is compressed to half of its length. If the force constant $k$ of spring is $6\text{N/m}$, then it's potential energy ($\text{joule}$) is:

1. $3x^2$
2. $\frac{2}{3}{x}^2$
   
3. $\frac{3}{4}{x}^2$
   
4. $6x^2$
   

Q. Consider the situation shown in figure. Initially, the spring is unstretched when the mass $m$ is released from rest. Assuming no friction in the pulley, find the maximum extension in the spring.

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

Q. A $3\text{kg}$ block collides with a massless spring of spring constant $110\text{N/m}$ attached to a wall. The speed of the block was observed to be $1.4\text{m/s}$ at the moment of collision. The acceleration due to gravity is $9.8{\text{m/s}}^2$. How far does the spring compress if the horizontal surface on which the mass moves is frictionless?

1. $0.23\text{m}$
2. $0.41\text{m}$
3. $2.5\text{m}$
4. $3.2\text{m}$

Q. A long spring is stretched by $2\text{cm}$. Its potential energy is $\text{U}$. If the spring is stretched by $10\text{cm}$, its potential energy would be

1. $\frac{U}{25}$
2. $\frac{U}{5}$
3. $5U$
4. $25U$

Q.

Figure shows a smooth curved track terminating in a smooth horizontal part. A spring of spring constant 400 N/m is attached at one end to a wedge fixed rigidly with the horizontal part. A 40 g mass is released from rest at a height of 4.9 m on the curved track. Find the maximum compression of the spring. (g = 9.8 m/$s^2$)

1. 4.9 m

2. 4.6 m

3. 9.8 m

4. None of these

Q. If $U_1,U_2,U_3$ represent the potential energy differences for moving a particle from $A$ to $B$ along three different paths $1,2\&3$ (as shown in the figure) in the gravitational field of point mass $m$, find the correct relation between $U_1,U_2\&U_3$.

1. $U_1=U_2=U_3$
2. $U_1>U_2>U_3$
3. $U_1<U_2<U_3$
4. $U_1>U_2<U_3$

Q. The length of a rod is $20\text{cm}$ and area of cross-section $2{\text{cm}}^2$. The Young's modulus of the material of wire is $1.4\times {10}^{11}{\text{N/m}}^2$. If the rod is compressed by a load of $5\text{kg-wt}$ along its length, then find the energy stored in the rod (in Joules).

1. $8.57\times {10}^{-6}$
2. $22.5\times {10}^{-4}$
3. $9.8\times {10}^{-5}$
4. $45.0\times {10}^{-5}$

Q. Figure shows a spring fixed at the bottom end of an incline of inclination ${37}^{\circ }$. A small block of mass $2\text{kg}$ starts slipping down the incline from a point $5.8\text{m}$ away from the spring. The block compresses the spring by $20\text{cm}$, stops momentarily and then rebounds through a distance of $1\text{m}$ up the incline. The friction coefficient between the plane and the block and the spring constant of the spring is (Take $g=10{\text{m/s}}^2$)

1. $\mu =0.54,k=1032\text{N/m}$
2. $\mu =0.12,k=1032\text{N/m}$
3. $\mu =0.12,k=962\text{N/m}$
4. $\mu =0.28,k=962\text{N/m}$

Q. Two wires of the young's modulii $Y$ and $2Y$ having lengths $2L,L$ and radii $2R,R$ respectively, are joined end to end as shown in the image. The elastic potential energy stored in the system in equilibrium, is


[Assume the wires are massless and $w$ is the weight hung at the bottom]

1. $\frac{4w^{2}L}{\pi R^{2}Y}$
2. $\frac{w^{2}L}{4\pi R^{2}Y}$
3. $\frac{2w^{2}L}{\pi R^{2}Y}$
4. $\frac{w^{2}L}{2\pi R^{2}Y}$

Q. A spring of unstretched length $l$ and force constant $k$ is stretched by a small length $x$. It is further stretched by another small length $y$. The work done in the second stretching is

1. $\frac{1}{2}k{x}^2$
2. $\frac{1}{2}k(x^2+y^2)$
3. $\frac{1}{2}ky(2x+y)$
4. $\frac{1}{2}kx(x+2y)$

Q. A spring block system is compressed $x\text{m}$ from the mean position and released. Find out the total work done by the spring force when it reaches the other extreme point (fully stretched). Neglect friction.

1. $\frac{1}{2}k{x}^2$
2. $k{x}^2$
3. $\frac{1}{2}k(2x{)}^2$
4. $0$

Q. An ideal spring with spring constant $k$ is hung from the ceiling and a block of mass $m$ is attached to its lower end. The mass is released with the spring initially unstretched. The maximum extension in the spring is

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