The Forces of Rolling — Question 6

Question 6

Insufficient static friction invalidates a downhill rolling solution

A hoop starts from rest on a $30^\circ$ incline with $\mu_s=0.20$ and $\mu_k=0.15$. Use $I=MR^2$ and $g=9.8$.

1. Test whether pure rolling is possible.

2. If it slips downhill, calculate center acceleration and $R|\alpha|$.

Question 6 - Solution

Insufficient static friction invalidates a downhill rolling solution

(a) Derivation and calculation

For a hoop, the candidate rolling acceleration and friction are $$a_{\mathrm{roll}}=g\sin\theta/2,\qquad f_{\mathrm{roll}}=Mg\sin\theta/2.$$ Therefore $$\mu_{s,\mathrm{required}}=\tfrac12\tan30^\circ=.2887>.20.$$ The no-slip solution fails its static-friction test.

(b) Result and interpretation

Use uphill kinetic friction $f_k=\mu_kMg\cos\theta$ instead: $$\boxed{a=g(\sin30^\circ-.15\cos30^\circ)=3.627\ \mathrm{m/s^2}},$$ $$\boxed{R|\alpha|=\frac{f_kR^2}{I}=.15g\cos30^\circ=1.273\ \mathrm{m/s^2}}.$$ The contact-slip acceleration is $a-R|\alpha|=2.354$ m/s$^2$ downhill, confirming the assumed friction direction.

Physical check

Once slipping begins, the rolling constraint cannot be imposed along with the kinetic-friction equations.