On a particle placed at origin a variable force $\mathrm{F}=-\mathrm{ax}$ (where a is a positive constant) is applied. If $\mathrm{U}\left(0\right)=0$, the graph between potential energy of particle U(x) and x is best represented by

1. 

2. 

3. 

4. 

The variation of potential energy U of a body moving along x-axis varies with its position (x) as shown in figure

The body is in equilibrium state at

1. A

2. B

3. C

4. Both A & C

A uniform chain of length L and mass M is lying on a smooth table and one third of its length is hanging vertically down over the edge of the table. If g is acceleration due to gravity, the minimum work required to pull the hanging part of the chain on the table is

1. MgL

2. $\frac{\mathrm{MgL}}{3}$

3. $\frac{\mathrm{MgL}}{9}$

4. $\frac{\mathrm{MgL}}{18}$

A block of mass m moving with velocity v₀ on a smooth horizontal surface hits the spring of constant k as shown. The maximum compression in spring is

1. $\sqrt{\frac{2\mathrm{m}}{\mathrm{k}}}.v_0$

2. $\sqrt{\frac{\mathrm{m}}{\mathrm{k}}}.v_0$

3. $\sqrt{\frac{\mathrm{m}}{2\mathrm{k}}}.v_0$

4. $\frac{\mathrm{m}}{2\mathrm{k}}.v_0$

For a particle moving under the action of a variable force, kinetic energy-position graph is given, then

1. At A particle is decelerating

2. At B particle is accelerating

3. At C particle has maximum velocity

4. At D particle has maximum acceleration

A particle moves with the velocity $\overrightarrow{\mathrm{v}}=(5\hat{\mathrm{i}}+2\hat{\mathrm{j}}-\hat{\mathrm{k}}){\mathrm{ms}}^{-1}$ under the influence of a constant force, $\overrightarrow{\mathrm{F}}(2\hat{\mathrm{i}}+5\hat{\mathrm{j}}-10\hat{\mathrm{k}})$ N. The instantaneous power applied is

1. 5 W

2. 10 W

3. 20 W

4. 30 W

A body is projected from ground obliquely. During downward motion, power delivered by gravity to it

1. Increases

2. Decreases

3. Remains constant

4. First decreases and then becomes constant

A body of mass m accelerates uniformly from rest to velocity v₁ in time interval T₁. The instantaneous power delivered to the body as a function of time t is

1. $\frac{{\mathrm{mv}}_1^2}{{\mathrm{T}}_1^2}\mathrm{t}$

2. $\frac{{\mathrm{mv}}_1}{{\mathrm{T}}_1^2}\mathrm{t}$

3. ${\left(\frac{{\mathrm{mv}}_1}{\mathrm{T}}\right)}^2\mathrm{t}$

4. $\frac{{\mathrm{mv}}_1^2}{{\mathrm{T}}_1}{\mathrm{t}}^2$

The power of a pump, which can pump 500 kg of water to height 100 m in 10 s is

1. 75 kW

2. 25 kW

3. 50 kW

4. 500 kW

A U-238 nucleus originally at rest, decays by emitting an $\mathrm{\alpha }$-particle, say with a velocity of v m/s. The recoil velocity (in ms^-1) of the residual nucleus is

1. $\frac{4\mathrm{v}}{238}$

2. $-\frac{4\mathrm{v}}{238}$

3. $\frac{\mathrm{v}}{4}$

4. $-\frac{4\mathrm{v}}{234}$