Rotational Motion — Dynamics & Conservation (Medium)

Quick Recap — Rotational Dynamics & Conservation

• Rolling race: smaller $\dfrac{I}{mR^2}$ wins — solid sphere ($0.40$) $>$ disc/solid cylinder ($0.50$) $>$ hollow sphere ($0.67$) $>$ ring/hoop ($1.0$).
• Minimum friction to roll (incline): $\mu_{min}=\dfrac{I/mR^2}{1+I/mR^2}\tan\theta$; the friction force is $f=\dfrac{I/mR^2}{1+I/mR^2}\,mg\sin\theta$.
• Falling "yo-yo" (string over a disc/cylinder): $a=\dfrac{g}{1+I/mR^2}$, string tension $T=m(g-a)$.
• Rotational collisions: common $\omega=\dfrac{I_1\omega_1+I_2\omega_2}{I_1+I_2}$; kinetic energy is lost.
• Conservation on a string: $L=mr^2\omega$ constant, so pulling the radius to half makes $\omega$ four times larger.

Worked mini-example. A rod ($L=1.2$ m) pivoted at one end is released from horizontal. Energy: $\tfrac12 I\omega^2=Mg\tfrac{L}{2}$ with $I=\tfrac{ML^2}{3}$ gives $\omega=\sqrt{\dfrac{3g}{L}}=\sqrt{25}=5$ rad/s, so the tip moves at $\omega L=6$ m/s.