Rotational Motion

**Rigid Body:** A rigid body consists of a number of particles such that the distance between any pair of particles always remains constant.

**Moment of Inertia (Rotational Inertia) I: -**

1. Mass of body

2. Distribution of mass about the axis of rotation

3. Moment of inertia of a body should always be referred to as about a given axis, since it depends upon distribution of mass about that axis.

4. It does not depend upon the state of motion of rotating body. It is same whether the body is at rest, rotating slowly or rotating fast about the given axis. I=∑mr2

**Rotational Kinetic Energy: -**

**Radius Gyration: -** Radius of gyration of a body about a given axis is that distance, at which if whole of the mass of the body were concentrated, it would have same moment of inertia as that of body.

Again, Radius of gyration of a body about a given axis is defined as the square root of the mean of the squares of distances of various particles of the body from the axis of rotation.

**System of mass for many particle system:** -

**1. Perpendicular axes theorem: - **

**2. ****Parallel axes theorem: - **

**Moments of Inertia of Various Objects: -**

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**Moment of inertia of a ring about an axis passing through its center and perpendicular to its plane: -**

(a) About one of its diameters: -

(b) About a tangent

**Moment of inertia of a solid disc: -**

(a) About an axis passing through its center and perpendicular to its plane: -

(b) About one of its diameters: -

(c) About a tangent: -

(i) Tangent lying in the plane of disc: -

(ii) Tangent perpendicular to the plane of disc: -

**Moment of inertia of an annular disc: -**

**(a) About an axis passing through the center and perpendicular to the plane:-**

**(b) About any of its diameter: -**

**(c) About a tangent: -**

**Moment of Inertia (I) and Angular momentum (L): -**

**Law of conservation of angular momentum: -**The net angular momentum of an isolated system (no external torque), always remains constant.

**Motion of a point mass attached to a string would over a cylinder capable of rotating about its axis of symmetry: -**

**Motion of a body rolling down an inclined plane without slipping: - **

The maximum allowed angle for rolling without slipping.