First slide

Rolling motion

Question

A small object of uniform density rolls up a curved surface with an initial velocity v. It reaches up to a maximum height of $\frac{3 v^{2}}{4 g}$ with respect to the initial position. The object is:

Moderate

A

ring
B

solid sphere
C

hollow sphere
D

disc

Solution

$\begin{array}{l} \frac{1}{2} {mv}^{2} (1 + \frac{K^{2}}{R^{2}}) = mgh \Rightarrow \frac{1}{2} {mv}^{2} (1 + \frac{K^{2}}{R^{2}}) = mg (\frac{3 v^{2}}{4 g}) \\ \Rightarrow \frac{K^{2}}{R^{2}} = \frac{1}{2} \Rightarrow f o r a disc \end{array}$

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