Four identical rods are joined end to end to form a square. The mass of each rod is $M$. The moment of inertia of the square about the median line is

An isosceles triangular piece is cut from a square plate of side $l$. The piece is one-fourth of the square and mass of the remaining plate is $M$. The moment of inertia of the plate about an axis passing through $O$ and perpendicular to its plane is

A disc of radius R rolls on a horizontal ground with linear acceleration $a$ and angular acceleration $\alpha$ as shown in Figure. The magnitude of acceleration of point P as shown in the figure at an instant when its linear velocity is v and angular velocity is $\omega$ will be

Two thin discs, each of mass $M$ and radius $r$ metre, are attached as shown in figure, to form a rigid body. The rotational inertia of this body about an axis perpendicular to the plane of disc $B$ passing through its centre is

Two identical masses are connected to a horizontal thin (massless) rod as shown in the figure. When their distance from the pivot is $D$, a torque $\tau$ produces an angular acceleration of ${\alpha }_1$. The masses are now repositioned, so that they are $2D$ from the pivot. The same torque produces an angular acceleration ${\alpha }_2$ which is given by, 

From a complete ring of mass $\mathrm{M}$ and radius $\mathrm{R}$, a $30°$ sector is removed. What is the moment of inertia of the incomplete ring about an axis passing through the centre of the ring and perpendicular to the plane of the ring is