Rotational Motion

Rotational


ROTATION OF A RIGID BODY :

A body which does not undergo any change in shape or size by the application of the external force is called a rigid body. There is no perfect rigid body in the

universe.

If a rigid body is made to rotate about an axis, every particle in the body has same angular displacement and angular velocity.

(The axis is outside the body).

₀ is the initial angular velocity, the angular acceleration, '' angular displacement after't' seconds and '' is the final angular velocity.








Moment Of Inertia

Moment of inertia of a rigid body about an axis is defined as the sum of the product and the square of the distance of the particle from the axis of rotation.

That is, I =

Here, m is mass of particle and r is distance of particle from axis of rotation.

Note : Moment of inertia plays the same role in rotational motion as mass plays in linear motion. That is, it is a measure of inertia in rotational motion and is also

called rotational inertia.


Radius of Gyration:

Suppose a body of mass M has moment of inertia I about an axis. The radius of gyration , k, of the body about that axis is defined as

I = Mk²

That is, k is the distance of the point mass M from the axis of rotation such that this point mass has the same moment of inertia about that axis as the given body.


Theorem Of Parallel Axes :

According to this theorem, the moment of inertia of body about moment of inertia of body about any axis is equal to the moment of inertia of body about a parallel

axis through the centre of mass plus the product of the mass of the body and the square of the distance between the two axes.

I = I_CM + Md²


Theorem Of Perpendicular Axes:

The moment of inertia of a plane body about an axis perpendicular to its plane is equal to the sum of its moments of inertia about two mutually perpendicular axes

in its own plane intersecting at the point through which the perpendicular axis passes.

i.e.,

Rotational Kinetic Energy :



Where I is the moment of inertia of the body. From this expression it is clear that moment of inertia is the rotational analogous of mass.

Torque due to a force is moment of force measures the turning effect of the force about the axis of rotation, and given as

= force × perpendicular distance from the axis of rotates.

Its unit is Newton meter (Nm)


Angular Momentum:

Angular momentum of particle about a given axis is the moment of linear momentum of the particle about axis. It is equal to the product of linear distance of the

line of action of linear momentum from axis of rotation and linear momentum.

i.e.,

The magnitude of angular momentum is




Law Of Conservation of Angular Momentum:

if , then , which means that is constant. This leads to the law of conservation of angular momentum.

If the resultant external torque on a system is zero, its total angular momentum remains constant.

I = constant

This is the rotational analogue of the law of conservation of linear momentum.


Work done by a Torque :

It can now be easily shown that if a torque rotates a body through a small angle d then the work done is



and, therefore, the total work done in rotating the body from ₁to ₂is



For a constant torque this becomes W = t , where = ₂ -₁.


Power :

The rate at which work is done by a torque, i.e., power in rotational motion is



p = w


Work-Energy Theorem :

It can be easily shown using the equations ₂= that the work done by the net torque is equal to the change in rotational kinetic energy.



Relation between Angular Momentum and Angular Velocity for a Rigid Body :
We have as I is constant for a rigid body. Comparing this with




we get,

Which is the rotational analogue of p = mv.


Equilibrium of a Rigid Body:

(a) Translational Equilibrium. For a body to be in translational equilibrium, the vector sum of all the external forces on the body must be zero.

(b) Rotational Equilibrium. For a body to be in rotational equilibrium, the vector sum of all the external torques on the body about any axis must be zero.