Question

A small solid sphere of radius $r$ rolls down an incline without slipping which ends into a vertical loop of radius $R$. Find the height above the base so that it just loops the loop.

• A. $\frac{5}{2}R$
• B. $\frac{5}{2}(R-r)$
• C. $\frac{25}{10}(R-r)$
• D. $\frac{27}{10}R-\frac{17r}{10}$

Answer 1

The correct option is D $\frac{27}{10}R-\frac{17r}{10}$

From given,
The initial energy of the body will be equal to gravitational potential energy.
$E_i=mgh$

Now at final position point $f$:
potential energy is given by $E_f=mg(R-r)+\frac{1}{2}m{v}^2+\frac{1}{2}I{\omega }^2$ ...(1)

at point f:
$mg=\frac{m{v}^2}{R-r}$ ....(2)
using eq. (2) and (1), and q
$\Rightarrow \frac{27}{10}R-\frac{17r}{10}$