A simple pendulum is set up in a trolley which moves to the right with an acceleration a on a horizontal plane. Then the thread of the pendulum in the mean position makes an angle $\mathrm{\theta }$ with the vertical

(1) ${\tan }^{-1}\left(\frac{\mathrm{a}}{\mathrm{g}}\right)$ in the forward direction

(2) ${\tan }^{-1}\left(\frac{\mathrm{a}}{\mathrm{g}}\right)$ in the upward direction 

(3) ${\tan }^{-1}\left(\frac{\mathrm{a}}{\mathrm{g}}\right)$ in the backward direction

(4) ${\tan }^{-1}\left(\frac{g}{a}\right)$ in the forward directions 

A simple pendulum hangs from the roof of a train moving on horizontal rails. If the string is inclined towards the front of the train, then train is

1. Moving with constant velocity

2. In accelerated motion

3. In retarded motion

4. At rest

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 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.

A motorcycle is going on an overbridge of radius \(R\). The driver maintains a constant speed. The normal force on the motorcycle as it ascends the overbridge will be:
1. increases.
2. decreases.
3. remains the same.
4. fluctuates erratically.

Two blocks A and B of masses m & 2m respectively are held at rest such that the spring is in natural length. Find the accelerations of both the blocks just after release.

(1) $g\downarrow , g\downarrow$

(2) $\frac{g}{3}\downarrow , \frac{g}{3}\uparrow$

(3) (0, 0)

(4) $g\downarrow , 0$

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 \(\omega\). If the tensions in the threads are the same during motion, the distance of \(M\) from the axis is:
1. \(\frac{M l}{M + m}\)
2. \(\frac{m l}{M + m}\)
3. \(\frac{M+m}{M}l\)
4. \(\frac{M+m}{m}l\)

A ball of mass 0.25 kg attached to the end of a string of length 1.96 m is moving in a horizontal circle. The string will break if the tension is more than 25 N. What is the maximum speed with which the ball can be moved 

(1) 14 m/s

(2) 3 m/s

(3) 3.92 m/s

(4) 5 m/s

A point mass m is suspended from a light thread of length l, fixed at O, is whirled in a horizontal circle at 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)

A motor cycle driver doubles its velocity when he is having a turn. The force exerted outwardly will be

(1) Double

(2) Half

(3) 4 times

(4) $\frac{1}{4}$ times