Rolling Motion: No-Slip Constraints, Inclines, and Energy

All flashcards

No-slip rolling

The contact point is instantaneously at rest relative to the surface, linking translation and rotation.

Rolling acceleration constraint

For constant radius and no slipping, the magnitudes satisfy $a_{CM}=R\alpha$.

Acceleration of a rolling object on an incline

With no slipping, $a_{CM}=\frac{mg\sin\theta}{m+I_{CM}/r^2}$.

Kinetic friction during slipping

Its magnitude is $f_k=\mu_kN$, and it opposes relative sliding at the contact.

Mechanical energy of a rolling object

It includes translational kinetic, rotational kinetic, and gravitational potential energy.

Quiz questions

1. A solid cylinder rolls without slipping down an incline at angle $\theta$. What is its center-of-mass acceleration?Answer: $a_{CM}=\frac{2}{3}g\sin\theta$.
2. What minimum coefficient of static friction allows a solid cylinder to roll without slipping down an incline at angle $\theta$?Answer: $\mu_s\geq\frac{1}{3}\tan\theta$.
3. A cylinder slides down an incline at angle $\theta$ while kinetic friction acts up the slope. What is its translational acceleration?Answer: $a_{CM}=g(\sin\theta-\mu_k\cos\theta)$.
4. A solid cylinder of radius $r$ slips down an incline at angle $\theta$ under kinetic friction. What is its angular acceleration?Answer: $\alpha=\frac{2\mu_k g\cos\theta}{r}$.
5. An object with moment of inertia $I_{CM}=kmr^2$ starts from rest and rolls without slipping through a vertical drop $h$. What is its final center-of-mass speed, neglecting dissipative effects?Answer: $v_{CM}=\sqrt{\frac{2gh}{1+k}}$.