Systems of Part...

The general motion of a rigid body can be considered to be a combination of (i) a motion of its centre of mass about an axis, and (ii) its motion about an instantaneous axis passing through the centre of mass.
These axes need not be stationary. Consider, for example, a thin uniform disc welded (rigidly fixed) horizontally at its rim to a massless stick, as shown in the figure.
When the disc-stick system is rotated about the origin on a horizontal frictionless plane with angular speed co, the motion at any instant can be taken as a combination of (i) a rotation of the centre of mass of the disc about the z-axis, and (ii) a rotation of the disc through an instantaneous vertical axis passing through its centre of mass (as is seen from the changed orientation of points P and Q).
Both these motions have the same angular speed ω this case.

A small cone filled with water is revolved in a vertical circle of radius 4 m and the water does not fall down. What must be the maximum period of revolution?

Three point masses m₁, m₂ and m₃ are placed at the corners of a thin massless rectangular sheet (1.2m × 1.0m), as shown in Fig. The centre of mass of the system is located at the point with coordinates

A circular plate of uniform thickness has a diameter of 56 cm. A circular portion of diameter 42 cm is removed from one edge as shown in Fig. The centre of mass of the remaining portion from the centre of plate will be at

A ring of mass m and radius r rotates about an axis passing through its centre and perpendicular to its plane with angular velocity ω. Its K.E. is

AB and CD are two identical rods each of length l and mass m joined to form a cross. The moment of inertia of these two rods about a bisector (XY) of angle between the rods is

Three identical bricks of length l are placed on the top of each other as shown in Fig, so that the part of each overhangs the one below. Find in terms of l, the maximum value of the total overhung x.

In a metallic triangular sheet ABC, AB = BC = l. If M is mass of sheet what is its moment of inertia about AC, Fig ?

A rigid spherical body is spinning around an axis without any external torque. Due to change in temperature, the volume increases by 1%. Its angular speed will

A thin circular ring of mass M and radius R is rotating about its axis with a constant angular speed ω. Two blocks each of mass m are attached gently to opposite ends of a diameter of the ring. The angular speed of the ring will be