The potential energy of a \(1 ~\text{kg}\) particle free to move along the \(x\text-\)axis is given by \(U(x)=\left(\frac {x^4}{ 4}-\frac {x^2}{ 2}\right)~\text J.\) The total mechanical energy of the particle is \(2~\text J.\) Then the maximum speed (in \(\text{ms}^{-1}\)) will be:
1. \(\dfrac{3}{\sqrt{2}} \)
2. \(\sqrt{2}\)
3. \(\dfrac{1}{\sqrt{2}}\)
4.  \(2\)

The position-time \((x\text- t)\) graph of a particle of mass \(2\) kg is shown in the figure. Total work done on the particle from \(t=0\) to \(t=4\) s is:
                   
1. \(8\) J
2. \(4\) J
3. \(0\) J
4. can't be determined

A body of mass 'm' is released from the top of a fixed rough inclined plane as shown in the figure. If the frictional force has magnitude F, then the body will reach the bottom with a velocity: $(\mathrm{L}=\sqrt{2}\mathrm{h})$

1. \(u^{2} \sin^{2}\alpha\)

2. \(\dfrac{m u^{2} \cos^{2} \alpha}{2}\)

3. \(\dfrac{m u^{2}\sin^{2} \alpha}{2}\)

4. \(- \dfrac{m u^{2}\sin^{2} \alpha}{2}\)

A particle of mass \(m\) is attached to a string and is moving in a vertical circle. Tension in the string when the particle is at its highest and lowest point is \(T_1\) $\mathrm{and}$ \(T_2\) respectively. Here \(T_2-T_1\) is equal to: 

The potential energy of a particle varies with distance \(r\) as shown in the graph. The force acting on the particle is equal to zero at:

1. \(P\)
2. \(S\)
3. both \(Q\) and \(R\)
4. both \(P\) and \(S\)

A block of mass m is placed in an elevator moving down with an acceleration $\frac{\mathrm{g}}{3}$. The work done by the normal reaction on the block as the elevator moves down through a height h is:

1.  $\frac{-2\mathrm{mgh}}{3}$

2.  $\frac{-\mathrm{mgh}}{3}$

3.  $\frac{2\mathrm{mgh}}{3}$

4.  $\frac{\mathrm{mgh}}{3}$

A graph is plotted by taking kinetic energy along the \(\mathrm{y}\)-axis and speed along the \(\mathrm{x}\)-axis for a constant mass. The slope of the graph at an instant represents:

A person-1 stands on an elevator moving with an initial velocity of 'v' & upward acceleration 'a'. Another person-2 of the same mass m as person-1 is standing on the same elevator. The work done by the lift on the person-1 as observed by person-2 in time 't' is:

1.  $\mathrm{m}\left(\mathrm{g}\right)$ $+$ $\mathrm{a}\left(\mathrm{vt}\right)$ $+$ $\frac{1}{2}{\mathrm{at}}^2$

2.  $-\mathrm{mg}\left(\mathrm{vt}\right)$ $+$ $\frac{1}{2}{\mathrm{at}}^2$

3.  0

4.  $\mathrm{ma}\left(\mathrm{vt}\right)$ $+$ $\frac{1}{2}{\mathrm{at}}^2$

A weight 'mg' is suspended from a spring. The energy stored in the spring is U. The elongation in the spring is:

1.  $\frac{2\mathrm{U}}{\mathrm{mg}}$

2.  $\frac{\mathrm{U}}{\mathrm{mg}}$

3.  $\frac{\sqrt{2}\mathrm{U}}{\mathrm{mg}}$

4.  $\frac{\mathrm{U}}{\sqrt{2}\mathrm{mg}}$