A spinning skater pulls their arms in and speeds up, though nobody pushed them and no motor engaged.

Pull the weights in#

Angular momentum is $L = I\omega$. Pulling toward the axis exerts no torque, so $L$ cannot change. $I$ can, and $\omega$ has no choice:

$$I_1\omega_1 = I_2\omega_2 \quad\Longrightarrow\quad \frac{\omega_2}{\omega_1} = \frac{I_1}{I_2}$$

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The energy is not free#

With $L$ fixed, $K = \tfrac{1}{2}I\omega^2 = L^2/2I$, so shrinking $I$ raises the energy. Angular momentum is conserved because there is no external torque. Energy changes because there is external work: the weights are circling, and pulling them inward takes force through a distance.

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What comes next#

You made a platform spin nearly seven times faster with no torque at all, and found the energy that took. Next, the same conservation law applied to a planet, where it turns out to be the reason Kepler's second law is true.