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. $r\ge \frac{\mu g}{{\omega }^2}$

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

1. $30 {\mathrm{ms}}^{-2} \mathrm{downwards}$                                     
2. $4 {\mathrm{ms}}^{-2} \mathrm{upwards}$
3. $4 {\mathrm{ms}}^{-2} \mathrm{downwards}$                                       
4. $14 {\mathrm{ms}}^{-2} \mathrm{upwards}$

A body, under the action of a force $\overrightarrow{F}=6\hat{i}-8\hat{j}+10\hat{k},$ acquires an acceleration of 1$m{s}^{-2}.$ The mass of this body must be

(1) $20\sqrt{10}kg$                                   

(2) 10kg

(3) 20kg                                         

(4) $10\sqrt{2}kg$                                

Three forces acting on a body are shown in the figure. To have the resultant force only along the y-direction, the magnitude of the minimum additional force needed is 

1. 0.5N

2. 1.5N

3. $\frac{\sqrt{3}}{4}$ N

4. $\sqrt{3}$N

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 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 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. $\alpha <\frac{g}{\mu }$

The angle of banking at the turning of a road does not depend upon the

(a) Mass of vehicle

(b) Acceleration due to gravity

(c) The velocity of the vehicle

(d) Radius of the curved path

The slope of a smooth banked horizontal surface road is $p$. If the radius of the curve be $r$, the maximum velocity with which a car can negotiate the curve is given by

(a) $prg$

(b) $\sqrt{prg}$

(c) $p\mathit{/}rg$

(d) $\sqrt{p\mathit{/}rg}$