Relation between Work Done and PE

Q. Work done by a conservative force is equal to negative change in the potential energy of the system.

1. False
2. True

Q. The potential energy of a particle varies with position $x\text{(in m)}$ according to the relation $U\left(x\right)=(x^2-4x)\text{J}$. The point $x=2\text{m}$ is a point of

1. stable equilibrium
2. unstable equilibrium
3. neutral equilibrium
4. none of the above

Q. When you have to increase the potential energy of an object then

1. You have to do positive work against gravity
2. gravity has to do positive work against you
3. Gravity should not do work
4. Positive work has to be done against you

Q.

The bob of a pendulum is released from a horizontal position A as shown in the figure. The length of the pendulum is 2 m. If $10\%$ of the initial energy of the bob is dissipated as heat due to the friction of air, what would be the speed of the bob when it reaches the lowermost point B? Take $g=10m{s}^{-2}$.

1. $3m{s}^{-1}$

2. $4m{s}^{-1}$

3. $5m{s}^{-1}$

4. $6m{s}^{-1}$

Q. Consider a rain drop of mass $1\text{g}$ falling from a height of $2\text{km}$. It hits the ground with a speed of $50\text{m/s}$. The work done by resistive force of air is
$(g=10{\text{m/s}}^2)$

1. $-187.5\text{J}$
2. $43.75\text{J}$
3. $-18.75\text{J}$
4. $14.375\text{J}$

Q. A uniform chain of mass $m$ and length $L$ hangs on a table with two thirds of its length on the table. The work done by a person to put the hanging part back on the table will be

1. $\frac{mgL}{18}$
2. $\frac{mgL}{9}$
3. $\frac{mgL}{27}$
4. $mgL$

Q. An $F-x$ graph is shown below. Find the difference in initial and final potential energies from $x=0\text{m}$ to $x=3\text{m}$.

1. $5\text{J}$
2. $10\text{J}$
3. $15\text{J}$
4. $20\text{J}$

Q. A rod $OA$ of mass $m$ and length $L$ is rotating in a horizontal circle about point $O$ as shown in figure. $B$ is the mid-point of rod. If $T$ represents tension in the rod at any point and $U_o$ represents elastic strain energy stored in the rod $OA$, $U_B$ represents elastic strain energy stored in the rod $\left(OB\right)$ then choose correct alternative (s).

1. $T_o=\frac{4}{3}{T}_B$
2. $T_o=2T_B$
3. $U_o=\frac{16}{11}{U}_B$
4. $U_o=4U_B$

Q. The potential energy of a system increases if work is done

1. upon the system by a non conservative force.
2. on the system against a conservative force.
3. by the system against a non conservative force.
4. upon the system by a conservative force.

Q. A particle moves under the influence of a force F = CX from X = 0 to $X=X_1$. The work done in this process will be

1. $\frac{C{X}_1^2}{2}$
2. $C{X}_1^2$
3. $C{X}_1^3$
4. $0$

Q. A body of mass 2kg moves under the influence of a force. Its position x changes with time according to the relation $x=\frac{t^3}{3}$ where x is in meter and t in seconds. The work done by this force in first two seconds will be

1. 1600 Joule
2. 160 Joule
3. 16 Joule
4. 1.6 Joule

Q. There are two massless springs A and B of spring constant $K_A$ and $K_B$ respectively and $K_A>K_B$. If $W_A$ and $W_B$ be denoted as work done on A and work done on B respectively, then

1. If they are compressed to same distance, $W_A>W_B$
2. If they are compressed by same force (upto equilibrium state) $W_A<W_B$
3. If they are compressed by same distance, $W_A<W_B$
4. If they are compressed by same force (upto equilibrium state) $W_A>W_B$

Q. A body of mass m is accelerated to velocity v in time $t_1$. The work done by the force as a function of time t will be

1. $\frac{m{v}^2{t}^2}{2e^2}$
2. $\frac{m{v}^2{t}^2}{2t_1^2}$
3. $\frac{mv}{2t}{t}^2$
4. $\frac{mv{t}^2}{2t}$

Q. The $F-X$ curve is shown below. Find the change in potential energy when particle displaces from $X=0\text{m}$ to $X=6\text{m}$ (Assume SI units)

1. $15\text{J}$
2. $30\text{J}$
3. $60\text{J}$
4. $7.5\text{J}$

Q. A rod $OA$ of mass $m$ and length $L$ is rotating in a horizontal circle about point $O$ as shown in figure. $B$ is the mid-point of rod. If $T$ represents tension in the rod at any point and $U_o$ represents elastic strain energy stored in the rod $OA$, $U_B$ represents elastic strain energy stored in the rod $\left(OB\right)$ then choose correct alternative (s).

1. $T_o=\frac{4}{3}{T}_B$
2. $T_o=2T_B$
3. $U_o=\frac{16}{11}{U}_B$
4. $U_o=4U_B$

Q. A man places a chain (of mass ${}^{\prime }{m}^{\prime }$ and length ${}^{\prime }{L}^{\prime }$) on a table slowly. Initially the lower end of the chain just touches the table. The man drops the chain when half of the chain is in vertical position. Then work done by the man in this process is:

1. $-mg\frac{L}{2}$
2. $-\frac{mgL}{4}$
3. $-\frac{3mgL}{8}$
4. $-\frac{mgL}{8}$

Q. A particle, starting at the origin, experiences a variable force defined by F(x) = 3x 2 , causing it to move along the x-axis. How much work is done on the particle from its starting point to x = 5 ?

1. 100 J
2. 125 J
3. 150 J
4. 135 J

Q. A body of mass m is accelerated to velocity v in time $t_1$. The work done by the force as a function of time t will be

1. $\frac{m{v}^2{t}^2}{2e^2}$
2. $\frac{m{v}^2{t}^2}{2t_1^2}$
3. $\frac{mv}{2t}{t}^2$
4. $\frac{mv{t}^2}{2t}$

Q. Potential energy of a system of particles is $U=\frac{\alpha }{3r^3}-\frac{\beta }{2r^2}$, where $r$ is distance between the particles. Here $\alpha$ and $\beta$ are positive constants. Which of the following are correct for the system?

1. Equilibrium separation between the particles is $\frac{\alpha }{\beta }$
2. For $r=\frac{\alpha }{\beta }$, the equilibrium is unstable
3. For $r=\frac{\alpha }{\beta }$, the equilibrium is neutral.
4. Work required to slowly move the particles to infinite separation from initial equilibrium position is $\frac{\beta }{\alpha }$

Q. A man is drawing water from a well of depth 10 m using a bucket with hole in it, such that only half of the water remains when bucket arrives at top of the well.
When bucket is full of water, its mass is 40 kg and pulling rate of bucket is constant. The work done by man in kilo joules to pull bucket up is ( g = 10 m/ $s^2$) [ Neglect weight of the bucket.]

Q. A body of mass $5\text{kg}$ is dropped from a tower of height $10\text{m}$. Find the difference between work done $\text{(}J)$ by gravity and change in potential energy of the body from ${}^{\prime }{A}^{\prime }$ to ${}^{\prime }{B}^{\prime }$ (Take $g=10\frac{m}{s^2}$)

1. $-1000\text{joule}$
2. $-100\text{joule}$
3. $1000\text{joule}$
   
4. $2000\text{joule}$

Q.

The bob of a pendulum is released from a horizontal position A as shown in the figure. The length of the pendulum is 2 m. If $10\%$ of the initial energy of the bob is dissipated as heat due to the friction of air, what would be the speed of the bob when it reaches the lowermost point B? Take $g=10m{s}^{-2}$.

1. $3m{s}^{-1}$

2. $4m{s}^{-1}$

3. $5m{s}^{-1}$

4. $6m{s}^{-1}$