Exploring torque, inertia distribution, and the conservation of angular momentum in rotating systems.
Just as force causes linear acceleration, torque ($\tau$) is the rotational analog of force that causes an object to undergo angular acceleration ($\alpha$). The relationship is governed by Newton's Second Law for rotation.
Where $\tau$ is the net applied torque, $I$ is the moment of inertia, and $\alpha$ is the angular acceleration. Under constant torque, the angular speed ($\omega$) increases linearly with time: $\omega = \omega_0 + \alpha t$.
Applying force at the outer edge of a merry-go-round yields a much larger torque than pushing close to the central spindle, making it much easier to accelerate.
Unlike translational mass, the rotational resistance of an object (the moment of inertia, $I$) depends not just on its total mass, but fundamentally on how that mass is distributed relative to the axis of rotation.
Mass located farther from the rotational axis contributes quadratically more to the moment of inertia. For a fixed mass, concentrating it at the rim makes it significantly harder to accelerate than concentrating it at the center.
The moment of inertia about any axis parallel to an axis through the center of mass is given by $I = I_{cm} + Md^2$, where $d$ is the distance between the two axes.
When no external net torque acts on a system, its total angular momentum ($L$) remains constant. This is the rotational analog to the conservation of linear momentum.
If the moment of inertia ($I$) is decreased by drawing mass inward (reducing the distribution radius $r$), the angular velocity ($\omega$) must increase proportionally to conserve $L$.
An ice skater spins slowly with arms outstretched. By drawing their arms and legs in close to their body, they decrease their moment of inertia and instantly spin much faster.