A thin wire of length $\ell$ and mass $m$ is bent in the form of a semicircle as shown. Its moment of inertia about an axis joining its free ends will be ...........

Four particles of masses $m_1 = 2m, m_2 = 4m, m_3 = m$ and $m_4$ are placed at four corners of a square. What should be the value of $m_4$ so that the centres of mass of all the four particles are exactly at the centre of the square ?

Two points of a rigid body are moving as shown. The angular velocity of the body is: ?

In the figure shown a ring $A$ is initially rolling without sliding with a velocity $v$ on the horizontal surface of the body $B$ (of same mass as $A$). All surfaces are smooth. $B$ has no initial velocity. What will be the maximum height reached by $A$ on $B$.

One end of a rod of length $L=1 \,m$ is fixed to a point on the circumference of a wheel of radius $R=1 / \sqrt{3} \,m$. The other end is sliding freely along a straight channel passing through the centre $O$ of the wheel as shown in the figure below. The wheel is rotating with a constant angular velocity $\omega$ about $O$. The speed of the sliding end $P$, when $\theta=60^{\circ}$ is

The moment of inertia of a thin circular lamina of mass $1\,kg$ and diameter $0.2\,metre$ rotating about one of its diameter is