Five persons \(A,\ B,\ C,\ D\ \text{and}\ E \) are pulling a cart of mass \(100\ \text{kg}\) on a smooth surface, and the cart is moving with an acceleration \(3 \ \text{m /s} ^ 2\) in the East direction. When person \(A\) stops pulling, it moves with an acceleration \(1 \ \text{m /s}^ 2\) in the west direction. When person\(​​B\) stops pulling, it moves with an acceleration \(24 \ \text{m/s}^ 2\) in the north direction. The magnitude of the acceleration of the cart when only \(A\) and\(​​B\) pull the cart, keeping their directions the same as the old directions, is:

1. \(26 \ \text{m/s}^ 2\)
2. \(3\sqrt{71} \ \text{m/s}^ 2\) $\mathrm{}$
3. \(25 \ \text{m/s}^ 2\)
4. \(30 \ \text{m/s}^ 2\)

A body moves along an uneven surface with constant speed at all points. The normal reaction due to ground on the body is:

The force required to just move a body up an inclined plane is double the force required to just prevent it from sliding down. If \(\phi\) is the angle of friction and \(\theta\) is the angle which the plane makes with the horizontal, then:
1. \(\tan 𝜃=\tan \phi\)

2. \(\tan 𝜃=2\tan \phi\)$$

3. \(\tan 𝜃=3\tan \phi\)$$

4. \(\tan \phi=\tan \theta\)$$

If the overbridge is concave instead of being convex, the thrust on the road at the lowest position will be:
1. \(mg+\dfrac{mv^2}{r}\)
2. \(mg-\dfrac{mv^2}{r}\)
3. \(\dfrac{mv^2g}{r}\)
4. \(\dfrac{v^2g}{r}\)

A particle moves in a circular orbit under the action of a central attractive force inversely proportional to the distance \(r\). The speed of the particle is:

1. Proportional to \(r^2\)

2. Independent of \(r\)

3. Proportional to \(r\)

4. Proportional to \(\dfrac 1r\)

Two masses \(M\) and \(m\) are attached to a vertical axis by weightless threads of combined length \(l\). They are set in rotational motion in a horizontal plane about this axis with constant angular velocity \(ω\). If the tensions in the threads are the same during motion, the distance \(x\) of \(M\) from the axis is:

1. \(\dfrac{Ml}{M+m}\)

2. \(\dfrac{ml}{M+m}\)

3. \(\dfrac{M+m}{M}l\)

4. \(\dfrac{M+m}{m}l\)

A point mass \(m\) is suspended from a light thread of length \(l,\) fixed at \(O\), and is whirled in a horizontal circle at a constant speed as shown. From your point of view, stationary with respect to the mass, the forces on the mass are:

1.		2.
3.		4.

Three identical particles are joined together by a thread as shown in the given figure. All three particles are moving in horizontal circles centred at \(O\). If the velocity of the outermost particle is \(v_0\), then the ratio of tensions in the three sections of the string is:

1. \(3 : 5 : 7\)
2. \(3 : 4 : 5\)
3. \(7 : 11 : 6\)
4. \(3 : 5 : 6\)

A bucket full of water is revolved in vertical circle of radius 2m. What should be the maximum time-period of revolution so that the water doesn't fall off the bucket ?

1. 1 sec

2. 2 sec

3. 3 sec

4. 4 sec

A tube of length L is filled completely with an incompressible liquid of mass M and closed at both the ends. The tube is then rotated in a horizontal plane about one of its ends with a uniform angular velocity ω. The force exerted by the liquid at the other end is 

1. $\frac{ML{\omega }^2}{2}$

2. $ML{\omega }^2$

3. $\frac{ML{\omega }^2}{4}$

4. $\frac{M{L}^2{\omega }^2}{2}$