18.4 Rotation of rigid body  (Page 2/4)

$$\begin{array}{l}v=\omega r\end{array}$$

where "r" is the perpendicular distance of the particle's position from the axis of rotation in the plane of rotation. It is easy to visualize that particles constituting rigid body at different levels undergo rotation in different planes of rotation.

Pure rotational motion

It is also clear from the given relation for linear velocity that particle closer to the axis (smaller "r") will have lesser linear velocity than the one away from the axis (greater "r").

However, each of the particle of the body undergoes same angular displacement (θ) and has same angular velocity (ω) and angular acceleration (α). In other words, we can say that the angular attributes of motion of the rigid body in rotation are uniquely (single valued) defined at a particular instant.

This situation is analogous to pure translational motion of rigid body, in which each particle constituting the body has same linear velocity and acceleration at a particular instant. For this reason, the motion of a particle or a rigid body in pure rotational motion is governed by the same form of Newton's second law i.e. the relation that connects external torque to the angular acceleration has the same form as that of translational motion. In translational motion, this relationship is given as :

$$\begin{array}{l}\mathbf{F}=m\mathbf{a}\end{array}$$

Intuitively, we may conclude that relation between torque (cause) and angular acceleration (effect) might have the following form :

However, we find that the role of inertia is not represented by mass alone in rotation. As there are corresponding quantities for force and acceleration, there is also a separate corresponding term or quantity that represents inertia to rotation. This quantity is termed as "moment of inertia", represented by symbol, "I". The equivalent Newton's second law for rotation is, thus :

However, we need to evaluate moment of inertia of the rigid body appropriately so that it represents the inertia of the body to external torque (cause). In the next section, we shall drive the basic expression of moment of inertia first for a single particle, then for a system of particle and then finally for the rigid body.

Moment of inertia of a particle

For the rotation of a particle, we simulate a situation in which a particle mass "m" is rotated by applying external force. We consider a particle of mass "m" is attached to one end of a mass-less (it is a mere theoretical consideration) rod of length "r". The other end of the rod is hinged at "O" such that the particle can be rotated in a horizontal plane about a vertical axis passing through center of circle, "O" .

Rotation

Now let us consider that a force " F " is applied to the particle in the plane of the circular path as shown in the figure. The radial component of force ( $F_R$ ) does not produce any acceleration as the line of action of the radial force passes through the axis of rotation. The tangential component of force ( $F_T$ ) produces tangential acceleration, $a_T$ . The particle follows a circular path of radius "r" with its center at "O". From second law of translational motion,