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

A car of mass 1000 kg negotiates a banked curve of radius 90m on a frictionless road. If the banking angle is $45°$,the speed of the car is 

1. 20 $m{s}^{-1}$                                             2. 30 $m{s}^{-1}$

3. 5 $m{s}^{-1}$                                               4. 10 $m{s}^{-1}$

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:

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 body of mass M hits normally a rigid wall with velocity v and bounces back with the same velocity. The impulse experienced by the body is

(1) 1.5 Mv                                                 

(2) 2 Mv

(3) zero                                                     

(4) Mv

A conveyor belt is moving at a constant speed of 2 m/s. A box is gently dropped on it. The coefficient of friction between them is $\mathrm{\mu }=0.5.$ The distance that the box will move relative to the belt before coming to rest on it taking $\mathrm{g}=10 {\mathrm{ms}}^{-2}$, is 

1. $1.2 \mathrm{m}$                                            2. $0.6 \mathrm{m}$

3. zero                                               4. $0.4 \mathrm{m}$

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 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:

(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 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) $10\sqrt{2}kg$