A particle moving with velocity $\overrightarrow{v}$ is acted by three forces shown by the vector triangle PQR. The velocity of the particle will:

1. change according to the smallest force $\overrightarrow{QR}$.

2. increase.

3. decrease.

4. remain constant.

A block of mass 10 kg is in contact with the inner wall of a hollow cylindrical drum of radius 1 m. The coefficient of friction between the block and the inner wall of the cylinder is 0.1. The minimum angular velocity needed for the cylinder, which is vertical and rotating about its axis, will be:
(g=10 m/s²)

1. $10 \mathrm{\pi } \mathrm{rad}/\mathrm{s}$

2. $\sqrt{10} \mathrm{\pi } \mathrm{rad}/\mathrm{s}$

3. $\frac{10}{2\mathrm{\pi }} \mathrm{rad}/\mathrm{s}$

4. $10 \mathrm{rad}/\mathrm{s}$

A truck is stationary and has a bob suspended by a light string in a frame attached to the truck. The truck suddenly moves to the right with an acceleration of a. In the frame of the truck, the pendulum will tilt:

1. to the left and angle of inclination of the pendulum with the vertical is ${\sin }^{-1}\left(\frac{a}{g}\right)$

2. to the left and angle of inclination of the pendulum with the vertical is ${\cos }^{-1}\left(\frac{a}{g}\right)$

3. to the left and angle of inclination of the pendulum with the vertical is ${\tan }^{-1}\left(\frac{a}{g}\right)$

4. to the left and angle of inclination of the pendulum with the vertical is ${\tan }^{-1}\left(\frac{g}{a}\right)$

A body of mass m is kept on a rough horizontal surface (coefficient of friction= $\mu$). A horizontal force is applied to the body, but it does not move. The resultant of normal reaction and the frictional force acting on the object is given by $\overrightarrow{F}$, where:

1. $\left|\overrightarrow{F}\right|=mg+\mu mg$

2. $\left|\overrightarrow{F}\right|=\mu mg$

3. $\left|\overrightarrow{F}\right|\le mg\sqrt{1+{\mu }^2}$

4. $\left|\overrightarrow{F}\right|=mg$

Two bodies of mass, 4 kg and 6 kg, are tied to the ends of a massless string. The string passes over a pulley, which is frictionless (see figure). The acceleration of the system in terms of acceleration due to gravity (g) is:

1.  g/2  

2.  g/5

3.  g/10

4.  g 

A point mass 'm' is moved in a vertical circle of radius 'r' with the help of a string. The velocity of the mass is $\sqrt{7\mathrm{gr}}$ at the lowest point. The tension in the string at the lowest point is:

1. 6mg
2. 7mg
3. 8mg
4. mg

Calculate the acceleration of the block and trolly system shown in the figure. The coefficient of kinetic friction between the trolly and the surface is 0.05.

( $g = 10 m/s^2,$ mass of the string is negligible and no other friction exists ). 

1. 1.25 $m /s^2$

2. 1.50  $m /s^2$

3. 1.66 $m /s^2$

4. 1.00 $m /s^2$

A ball of mass 0.15 kg is dropped from a height 10 m, strikes the ground and rebounds to the same height. The magnitude of impulse imparted to the ball is $\left(\mathrm{g}=10 \mathrm{m}/{\mathrm{s}}^2\right)$ nearly:

1. 2.1 kg m/s

2. 1.4 kg m/s

3. 0 kg m/s

4. 4.2 kg m/s

Assertion (A): A standing bus suddenly accelerates. If there was no friction between the feet of a passenger and the floor of the bus, the passenger would move back.
Reason (R): In the absence of friction, the floor of the bus would slip forward under the feet of the passenger.

1. (A) is true but (R) is false.
2. (A) is false but (R) is true.
3. Both (A) and (R) are true and (R) is the correct explanation of (A).
4. Both (A) and (R) are true but (R) is not the correct explanation of (A).