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How does moment of inertia affect angular velocity?

Last updated date: 24th Jul 2024
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Hint : We know the definitions of angular velocity and moment of inertia of a body, and also know how they are connected to angular momentum. Again, as we know the angular momentum of a body is always conserved, so, from their relation, we can easily find out how moment of inertia affects angular velocity.

Complete Step-by-step solution:
The rotational equivalent of linear momentum is called angular momentum. Angular momentum can be defined mathematically as, $L=I\omega$ , where,
$L$ = angular momentum,
$I$ = moment of inertia, and,
$\omega$= angular velocity of the body or system.
Moment of inertia – The moment of inertia of a particle of mass $m$ about a line or axis is defined as, $I=m{{r}^{2}}$ , where $r$ is the distance from mass to the center or axis of motion. This quantity is used to express the tendency of a body to defend angular acceleration.
Angular velocity – The quantity angular velocity refers to the speed at which an object rotates or revolves with respect to another point. It means, the angular velocity of an object changes over time. It can be mathematically defined as, $\omega =\dfrac{d\theta }{dt}$ , where $\theta$ is the angle of rotation.
Now, we know the principle of conservation of angular momentum, which can be stated as – The total angular momentum of a particle remains constant in time if the total (external) torque acting on the particle is zero.
So, from the relation, $L=I\omega$ , if $L$ is conserved, then $I$ and $\omega$ must be inversely proportional to each other. It means, if the moment of inertia of a body increases, angular velocity of that body must decrease, and if moment of inertia of a body is decreased, angular velocity of that body must increase.

Note:
Angular momentum of a body has an unit kilogram meters squared per second ( $kg{{m}^{2}}/\sec$ ), moment of inertia of a body has an SI unit kilogram meters squared ( $kg{{m}^{2}}$ ), and angular velocity of a body has the SI unit as ${{s}^{-1}}$ .