A mass \(m\) is attached to a thin wire and whirled in a vertical circle. The wire is most likely to break when:

When an object is shot from the bottom of a long, smooth inclined plane kept at an angle of \(60^\circ\) $$with horizontal, it can travel a distance \(x_1\) along the plane. But when the inclination is decreased to \(30^\circ\) $$and the same object is shot with the same velocity, it can travel \(x_2\) distance. Then \(x_1:x_2\) will be:
1. \(1:2\sqrt{3}\)
2. \(1:\sqrt{2}\)
3. \(\sqrt{2}:1\)
4. \(1:\sqrt{3}\)

Body \(\mathrm{A}\) of mass \(4m\) moving with speed \(u\) collides with another body \(\mathrm{B}\) of mass \(2m\) at rest. The collision is head-on and elastic in nature. After the collision, the fraction of energy lost by the colliding body \(\mathrm{A}\) is:
1. \(\frac{5}{9}\)
2. \(\frac{1}{9}\)
3. \(\frac{8}{9}\)
4. \(\frac{4}{9}\)

\(300 ~\text{J}\) of work is done in sliding a \(2~\text{kg}\) block up an inclined plane of height \(10~\text{m}\). Taking \(g=\) \(10\) m/s², work done against friction is:
1. \(200 ~\text{J}\)
2. \(100 ~\text{J}\)
3. \(\text{zero}\)
4. \(1000 ~\text{J}\)

A body of mass 3 kg is under a constant force which causes a displacement s in metres in it, given by the relation s = $\frac{1}{3}$ t², where t is in sec. Work done by the force in 2 sec is:

1. $\frac{5}{19}J$

2. $\frac{3}{8}J$

3. $\frac{8}{3}J$

4. $\frac{19}{5}J$

The potential energy of a long spring when stretched by \(2\) cm is \(U\). If the spring is stretched by \(8\) cm, the potential energy stored in it is:
1. \(4U\)
2. \(8U\)
3. \(16U\)
4. \(U/4\)

A vertical spring with a force constant \(k\) is fixed on a table. A ball of mass \(m\) at a height \(h\) above the free upper end of the spring falls vertically on the spring so that the spring is compressed by a distance \(d\). The net work done in the process is:
1. \(mg(h+d)+\frac{1}{2}kd^2\)
2. \(mg(h+d)-\frac{1}{2}kd^2\)
3. \(mg(h-d)-\frac{1}{2}kd^2\)
4. \(mg(h-d)+\frac{1}{2}kd^2\)

A shell of mass 200 g is ejected from a gun of mass 4 kg by an explosion that generates 1.05 kJ of energy. The initial velocity of the shell is:

1.  100 ms^-1

2.  80 ms^-1

3.  40 ms^-1

4.  120 ms^-1