First slide

Rolling motion

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

Difficult

A

$\frac{5}{2} R$
B

$\frac{5}{2} (R - r)$
C

$\frac{25}{10} (R - r)$
D

$\frac{27}{10} R - \frac{17 r}{10}$

Solution

The minimum velocity at P, top of the loop, should b $v = \sqrt{g (R - r)}$ if the sphere keeps on rolling at top $v = ω R$
$\begin{array}{l} m g h = \frac{1}{2} m v^{2} + \frac{1}{2} I ω^{2} + m g (2 R - r) \\ = \frac{1}{2} m g (R - r) + \frac{1}{2} (\frac{2}{5}) m R^{2} 2 ω^{2} + m g (2 R - r) \\ = \frac{7}{10} m g (R - r) + m g (2 R - r) [ω R = v = \sqrt{g (R - r)}] \\ = \frac{m g}{10} (27 R - 17 r) or h = \frac{1}{10} (27 R - 17 r) \end{array}$

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