Question

A sphere rolls without slipping on an incline of inclination $\theta$. The minimum coefficient of static friction to support pure rolling is to be:

• A. $\frac{2}{3}\tan \theta$
• B. $\frac{2}{5}\tan \theta$
• C. $\frac{2}{7}\tan \theta$
• D. $\frac{3}{7}\tan \theta$

Answer 1

The correct option is C $\frac{2}{7}\tan \theta$

For pure rolling

$a_{com}=\alpha ra=accelarationofCOM\alpha =radialaccelarationr=radius(gsin\theta -\frac{f}{m})=\frac{\tau }{I}rgsin\theta -\frac{\mu mgcos\theta }{m}=\frac{\left(\mu mgcos\theta \right)r}{\frac{2}{5}{r}^{2}m}\times r\left[torqueisonlyduetofriction\right]gsin\theta =\frac{5}{2}\mu gcos\theta +\mu gcos\theta =\frac{7}{2}\mu gcos\theta \mu =\frac{2}{7}tan\theta$