PHSX 220 Physics I (w/ Calculus) · Unit 3: Momentum, rotation, gravitation · on Exam 3
Ch 11 Rolling motion and angular momentum
Study notes for Rolling motion and angular momentum in PHSX 220 Physics I (w/ Calculus) at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in StudyCat.
Big ideas
- Rolling without slipping: $v_{cm} = R\omega$, $a_{cm} = R\alpha$. Total KE $= \tfrac12 mv_{cm}^2 + \tfrac12 I\omega^2$. Rolling down a ramp: $v = \sqrt{\dfrac{2gh}{1 + I/(mR^2)}}$, so a sphere ($\tfrac25$) beats a disk ($\tfrac12$) beats a hoop ($1$), independent of mass and radius.
- Angular momentum of a rigid body $L = I\omega$; of a particle $L = mvr\sin\theta$ ($= mvr$ for circular motion). Newton for rotation: $\sum\vec\tau = d\vec L/dt$.
- Conservation: with zero net external torque, $L$ is conserved. Ice skater pulling in arms: $I$ drops so $\omega$ rises ($I_i\omega_i = I_f\omega_f$); kinetic energy increases (the skater does work).
- Direction by the right-hand rule: curl fingers with the rotation, thumb gives $\vec\omega$ and $\vec L$.
Key formulas
Rolling
$$v_{cm} = R\omega,\qquad K = \tfrac12 mv_{cm}^2 + \tfrac12 I\omega^2,\qquad v_{\text{bottom}} = \sqrt{\frac{2gh}{1 + I/(mR^2)}}$$
Angular momentum
$$L = I\omega,\qquad L = mvr\sin\theta,\qquad \sum\tau = \frac{dL}{dt},\qquad I_i\omega_i = I_f\omega_f$$
Worked example
A skater spins at 2.0 rev/s with arms out ($I = 4.0\ \text{kg·m}^2$) and pulls them in ($I = 1.6\ \text{kg·m}^2$). Find the new spin rate and the change in KE.
Show the solution
$\omega_f = 4.0(2.0)/1.6 = 5.0$ rev/s. In rad/s: $\omega_i = 12.6$, $\omega_f = 31.4$. $K_i = \tfrac12(4)(12.6^2) = 316$ J, $K_f = \tfrac12(1.6)(31.4^2) = 790$ J: KE rises by 474 J, supplied by the skater's muscles.
Common mistakes
- Forgetting the translational KE of a rolling object (or the rotational part).
- Expecting kinetic energy to be conserved when angular momentum is.
- Using v = Rω for a slipping wheel.
Exam tipFor "which rolls down fastest" questions, only the shape factor $I/(mR^2)$ matters: smaller factor wins. Mass and radius cancel.
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