Question

A ball of radius $r$ lies at the bottom of a vertical ring of radius $R$, find the minimum velocity to be given so that the ball completes the loop rolling without slipping.

• A. $\sqrt{5g(R-r)}$
• B. $\sqrt{\frac{27}{10}g(R-r)}$
• C. $\sqrt{\frac{27}{7}g(R-r)}$
• D. $\sqrt{\frac{27}{5}g(R-r)}$

Answer 1

The correct option is C $\sqrt{\frac{27}{7}g(R-r)}$

Using conservation of energy

$\frac{1}{2}m{v}_{bott}^2+\frac{1}{2}I{\omega }_{bott}^2=mg2(R-r)+\frac{1}{2}m{v}_{top}^2+\frac{1}{2}I{\omega }_{top}^2$

$m{v}_{bott}^2+\frac{2}{5}m{v}_{bott}^2=4mg(R-r)+m{v}_{top}^2+\frac{2}{5}m{v}_{top}^2$

$\frac{7}{5}m{v}_{bott}^2=4mg(R-r)+\frac{7}{5}m{v}_{top}^2=4mg(R-r)+\frac{7}{5}m\left[g\right(R-r\left)\right]$

or

$v_{bott}=\sqrt{\frac{27}{7}g(R-r)}$