The law of inertia, when applied to rotating systems, is expressed in terms of angular momentum. Just as linear momentum describes the resistance of an object to changes in linear motion, angular momentum describes the resistance of a rotating object to changes in rotational motion.

The law of inertia for rotating systems states that:

_In the absence of any external torque, the total angular momentum of a closed system remains constant._

Mathematically, this can be expressed as:

$$\sum L = constant$$

where:

- \(\sum L\) represents the total angular momentum of the system

- The summation takes into account the angular momentum of all individual components within the system

This means that a rotating system will continue to rotate at a constant angular velocity and in the same direction unless an external torque is applied to it. The system resists any attempt to change its rotational motion, just as a stationary object resists any attempt to change its state of rest.

If an external torque is applied to the system, the total angular momentum will change at a rate proportional to the applied torque. The greater the torque, the faster the change in angular momentum. This concept forms the basis for various applications, such as motors, gyroscopes, and angular velocity sensors.