Question

A sphere of mass m and radius r rolls down an incline and acquires enough speed to traverse a loop of radius R. The sphere rolls without slipping throughout its motion. The minimum elevation h of the starting point, measured from the top of the loop is :

• A. $\frac{R-17r}{5}$
• B. $7R$
• C. $17r$
• D. $\frac{7R-17r}{10}$

Answer 1

The correct option is D $\frac{7R-17r}{10}$

At position B,
decrease in P.E$=$ increase in K.E.
$\Rightarrow mg(h+r)=\frac{1}{2}m{v}^2+\frac{1}{2}I{\omega }^2$ $=\frac{1}{2}m{v}^2+\frac{1}{2}\left(\frac{2}{5}m{r}^2\right){\omega }^2$
But, ${\omega }^2=\frac{v^2}{r^2}$
For looping $v=\sqrt{g(R-r)}$
$\Rightarrow g(h+r)=\frac{1}{2}g(R-r)+\frac{1}{5}g(R-r)$
$\Rightarrow 10(h+r)=5(R-r)+2(R-r)$
$\Rightarrow 10h=7R-17r$
$\Rightarrow h=\frac{7R-17r}{10}$