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 particle of mass m is projected with velocity v making an angle of ${45}^{°}$ with the horizontal. When the particle lands on the level ground the magnitude of the change in its momentum will be 

1.  2mv

2. mv/$\sqrt{2}$

3. mv$\sqrt{2}$

4. zero 

A roller coaster is designed such that riders experience "weightlessness" as they go round the top of a hill whose radius of curvature is 20m. The speed of the car at the top of the hill is between:

1. 14m/s and 15m/s

2. 15m/s and 16m/s

3. 16m/s and 17m/s

4. 13m/s and 14m/s

The mass of a lift is 2000 kg. When the tension in the supporting cable is 28000 N, its acceleration is:

(a) $30 {\mathrm{ms}}^{-2} \mathrm{downwards}$                                      (b) $4 {\mathrm{ms}}^{-2} \mathrm{upwards}$

(c) $4 {\mathrm{ms}}^{-2} \mathrm{downwards}$                                        (d) $14 {\mathrm{ms}}^{-2} \mathrm{upwards}$

A gramophone record is revolving with an angular velocity $\omega .$ A coin is placed at a distance r from the centre of the record. The static coefficient of friction is $\mu .$ The coin will revolve with the record if: 

1. $r=\mu g{\omega }^2$                                       2. $r<\frac{{\omega }^2}{\mu g}$

3. $r\le \frac{\mu g}{{\omega }^2}$                                         4. $r\ge \frac{\mu g}{{\omega }^2}$

A block of mass m is in contact with the cart C as shown in the figure.

The coefficient of static friction between the block and the cart is $\mu .$The acceleration $\alpha$ of the cart that will prevent the block from falling satisfies

1. $\alpha >\frac{mg}{\mu }$                                       2. $\alpha >\frac{g}{\mu m}$

3. $\alpha \ge \frac{g}{\mu }$                                          4. $\alpha <\frac{g}{\mu }$

A person of mass 60 kg is inside a lift of mass 940 kg and presses the button on control panel. The lift starts moving upwards with an acceleration $1.0$ $\mathrm{m}/{\mathrm{s}}^2$. If $\mathrm{g}=10$ $\mathrm{m}/{\mathrm{s}}^2$, the tension in the supporting cable is:

1. 9680 N

2. 11000 N

3. 1200 N

4. 8600 N

A car of mass \(m\) is moving on a level circular track of radius \(R\). If \(\mu_s\) represent the static friction between the road and tyres of the car, then the maximum speed of the car in circular motion is given by:

The upper half of an inclined plane of inclination θ is perfectly smooth while the lower half is rough. A block starting from rest at the top of the plane will again come to rest at the bottom if the coefficient of friction between the block and lower half of the plane is given by

1. μ=1/tanθ

2. μ=2/tanθ

3. μ=2tanθ

4. μ=tanθ