Rotational Motion — Advanced Multi-Step Practice (Hard)

Quick Recap — Advanced Rotation (Multi-step)

• Landing/embedding (angular-momentum conservation about the axis): $I_1\omega_1=(I_1+I_{added})\omega_2$; a bullet/putty adds $mr^2$. Linear momentum is not conserved when there is a pivot.
• Rod released from horizontal (pivot at end): initial $\alpha=\dfrac{3g}{2L}$, so the tip's initial acceleration is $\dfrac{3g}{2}>g$ — objects resting on the far end lift off.
• Spin-up to rolling on the ground: angular momentum about the contact point is conserved; a disc set spinning at $\omega_0$ ends at $\omega=\dfrac{\omega_0}{3}$.
• Rolling identity: measure $a$ on an incline, then $\dfrac{I}{mR^2}=\dfrac{g\sin\theta}{a}-1$.
• Rigid-body scaling: for fixed mass, $L=I\omega$ constant means shrinking the radius by $n$ raises $\omega$ by $n^2$.

Worked mini-example. A disc ($M=4$ kg) spins at $\omega$; a $2$ kg putty lands on its rim. Here $I_{disc}=\tfrac12MR^2=2R^2$ and the putty adds $mR^2=2R^2$, so $\omega'=\dfrac{2R^2}{2R^2+2R^2}\omega=\dfrac{\omega}{2}$.