First slide

Rolling motion

Question

A thick walled hollow sphere has outer radius R. It rolls down an inclined plane without slipping and its speed at the bottom is v. If the inclined plane is frictionless and the sphere slides down without rolling, its speed at the bottom is 5v/4. What is the radius of gyration of the sphere?

Moderate

A

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

$\frac{R}{2}$
C

$\frac{3 R}{4}$
D

$\frac{\sqrt{3} R}{4}$

Solution

$\begin{array}{l} Case 1 : m g h = \frac{1}{2} m v^{2} + \frac{1}{2} m k^{2} ω^{2} \\ Case 2 : m g h = \frac{1}{2} m \times {(\frac{5 v}{4})}^{2} \\ ∴ \frac{1}{2} m k^{2} \frac{v^{2}}{R^{2}} + \frac{1}{2} m v^{2} = \frac{1}{2} m \times \frac{25 v^{2}}{16} \\ \Rightarrow \frac{k^{2}}{R^{2}} + 1 = \frac{25}{16} \\ \Rightarrow k = \frac{3 R}{4} \end{array}$

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