A small block of mass \((m=1\ \text{kg})\) is placed on the smooth inclined surface of a triangular wedge of mass \((M'=4\ \text{kg})\). The wedge rests on a smooth horizontal surface. A horizontal string connects the small block to a hanging block of mass \(M\) through a fixed, frictionless pulley, as shown in the figure.
If the inclination of the wedge is \(\theta=37^\circ\), find the value of \(M\) for which the small block does not slip relative to the wedge. Assume the string is light and inextensible.

           
1. \(12\) kg
2. \(15\) kg
3. \(10\) kg
4. \(4\) kg

A lighter body and a heavier body are both moving with the same momentum and are subjected to the same retarding force. Their stopping distances are \(s_1\) and \(s_2\) respectively. Then the correct relation between ${}_{}$ \(s_1\) and \(s_2\)  is:

1. \(s_1 >s_2\)

2. \(s_1<s_2\)

3. \(s_1=s_2\)

4. None of these

A simple pendulum (bob of mass \(m\)) oscillates in a vertical plane. Let \(\theta\) be the angular displacement from the vertical. The tension in the string is equal to \(mg \cos\theta\) under which condition?
1. Always
2. Never
3. At the extreme position
4. At the mean position

The graph for stopping time \((t)\) versus mass \((m)\) is shown in the figure. Then which one of the pairs is given?

1. Momentum, force

2. K.E., force

3. Velocity, force

4. None of these

A metal ring of mass \(m\) and radius \(R\) is placed on a smooth horizontal table and is set to rotate about its own axis in such a way that each part of the ring moves with a speed \(v\).

The tension in the ring is:

1. \(\dfrac{mv^2}{R}\)
2. \(\dfrac{mv^2}{2R}\)
3. \(\dfrac{mv^2}{2\pi R}\)
4. \(\text{None of these}\)

Tension on the string at point \(P\) is \(T\). The graph for tension \((T)\) versus \(x\) is shown in the figure. Then the string is:

While walking on ice, one should take small steps to avoid slipping. This is because smaller steps ensure:

1. Larger friction
2. Smaller friction
3. Larger normal force
4. Smaller normal force

A block is kept on a rough inclined plane with an angle of inclination \(𝜃\)$$. The graph of the net reaction (\(R\)) versus \(𝜃\) is:

1.		2.
3.		4.	None of these

A block is placed on a rough horizontal plane. A time-dependent horizontal force \(F=kt\) acts on the block. The acceleration-time graph of the block is :

1.		2.
3.		4.