18.4 Rotation of rigid body

Rotation of a rigid body is characterized by same angular velocity and acceleration of particles comprising it. The situation is similar to the case of translation in which linear velocity and acceleration of all particles comprising rigid body are same. In the previous module titled Rotation , we discussed torque as the “cause” of rotation and ways to calculate torque. In this module, we seek to study the torque (cause) and angular acceleration (effect) relationship for the rotational motion of a rigid body. In other words, we seek to state Newton’s laws of motion for rotation in line with the one that exists for translation.

Rigid body is composed of particles, which are at fixed distance with respect to each other. In simple words, if a particle "A" is at a distance of 10 mm (say) from another particle "B" within a rigid body, then they continue to remain 10 mm apart during motion. This requirement is important in describing rotational motion of a rigid body. The distribution of mass about the axis affects rotational inertia of the body. As such, change in inter-particle distance shall amount to changing "rotational inertia" of the body.

Before, we proceed we need to distinguish between two separate force requirements for rotational motion. In the previous module, we have discussed the force requirement for the torque which produces angular acceleration or causes rotational motion. What about the centripetal force requirement for a particle of the rigid body to move in circular motion? This force requirement is met by the inter-molecular forces. The requirement of centripetal force is the inherent requirement for circular motion of a particle and thereby for the rotation of rigid body. While studying cause and effect relation for the rotation, it should be clearly understood that we are only concerned with the force requirement of torque for the angular acceleration of the rigid body in rotation.

Newton's first law of rotation

In translation, a particle or particle like rigid body has constant linear velocity unless there is an external force being applied on it. By conjecture, we can extend this law to rotation saying that a rigid body in rotation about a fixed axis has constant angular velocity unless it is subjected to external torque. This is exactly the Newton's first law of rotation.

If the rigid body is at rest, then it will remain in rest. This is the exactly same assertion as for translation. On the other hand, if the rigid body is in rotation with a constant angular velocity, then it will continue to rotate with that angular velocity indefinitely. Of course, we do not realize the second assertion in our daily life because it is almost impossible to get rid of torques opposing rotational motion due to air resistance and resistance caused by the friction at the axis of rotation.

Newton's second law of rotation

Every particle of the rigid body in rotation undergoes circular motion irrespective of the shape of rigid body. The centers of the circular paths described by them lie on the axis of rotation. It should be noted that the different particles, constituting rigid body, have different linear velocities, but same angular velocity. It means that each particle traverses same angle in a given time. The linear velocity of a particle is related to angular velocity as :