The integral of torque with respect to time is equal to what?

The integral of torque with respect to time is equal to the change in angular momentum. This relationship arises from the fundamental definitions in dynamics and angular motion.

Torque can be defined as the rate of change of angular momentum. Mathematically, this is expressed by the equation:

[

\tau = \frac{dL}{dt}

]

where ( \tau ) represents torque and ( L ) is angular momentum. If we integrate torque with respect to time, we can derive the change in angular momentum over that time period:

[

\int \tau , dt = \Delta L

]

This implies that the total torque applied to an object over a given time interval results in a corresponding change in angular momentum. Thus, evaluating the integral effectively provides the difference in the object's angular momentum before and after the application of the torque.

Understanding this principle is essential for analyzing rotational dynamics, as it connects the concepts of force, motion, and energy in angular contexts.