System Of Particles Rotational Motion

In rolling without slipping acceleration of all the objects down the plane is different. Acceleration of rolling body down an inclined plane

For solid sphere;


For hollow sphere;


For disc;


It is obvious that acceleration of solid sphere is maximum, so solid sphere will reach the ground in least time.

Torque is the rate of change of angular momentum. That is,
If torque is zero, then
or constant
i.e., angular momentum of the system is conserved.

Let rod is placed along axis. Mass of element PQ of length
situated at is

The COM of the element has coordinates Therefore, coordinate of COM of the rod will be

In pure rolling, mechanical energy remains conserved. Therefore, when heights of inclines are equal, speed of sphere will be same in both the cases. But as acceleration down the plane, therefore, acceleration and time of descent will be different.

MI of disc about tangent in a place

MI of disc about tangent perpendicular to plane