Consider a mass m attached to the end of a massless rod as shown in Fig. 5.8.let us assume that the bearing at the pivot point O is frictionless. Let the system be in a horizontal plane. A force F is acting on the mass perpendicular to the rod and hence, this will accelerate the mass according to
F = ma
So, F = mra
As turning effect is produced by torque τ, it would, therefore, be better to write the equation for rotation in terms of torque. This can be done by multiplying both sides of the above equation by r. Thus
Which is rotational analogue of the Newton’s second law of motion, F = ma.
Do You Know?
Here F is replaced by τ, a by a and m by mr2. The quantity mr2 is known as the moment of inertia and is represented by I. the moment of inertia plays the same role in angular motion as the mass in linear motion. It may be note that moment of inertia depends not only on mass m but also on r2.most rigid bodies have different mass concentration at different distances from the axis of rotation, which means the mass distribution is not uniform. As shown in Fig. 5.9(a), the rigid body is made up of n small pieces of masses
Since the body is rigid, so all the masses are rotating with the same angular acceleration a,
Where I is the moment of inertia of the body and is expressed as