A plank is rotating in a vertical plane about one of its ends with a constant angular velocity $\omega =\sqrt{2}$ rad/s. A block of mass m = 2 kg is placed at a distance I = 1m from its end A (see diagram) which is hinged. The block starts sliding down when the plank makes an angle $\theta =30^{\circ }$ with the horizontal. If coefficient of friction between the plank and the block is $\mu$ and given that ${\mu }^2=\text{k/25}$. The value of k is

Two masses $m$ and $M$ are attached with strings as shown in the figure. For the system to be in equilibrium, we have

A uniform rope of length $L$ and mass $M$ is placed on a smooth fixed wedge as shown. Both ends of rope are at same horizontal level. The rope is initially released from rest, then the magnitude of initial acceleration of rope is

A bob is hanging over a pulley inside a car through a string. The second end of the string is in the hand of a person standing in the car. The car is moving with constant acceleration $a$ directed horizontally as shown in figure. The other end of the string is pulled with constant acceleration $a$ (relative to car) vertically. The tension in the string is equal to,

Statement I : If there is no external torque on a body about its centre of mass, then the velocity of the centre of mass remains constant.

Statement II : The linear momentum of an isolated system remains constant.

Select the correct option.