First slide

Dynamics of rotational motion about fixed axis

Question

A solid cylinder of mass M and radius R without slipping down an inclined plane of length L and height h. What is the speed of centre of mass when the cylinder reaches bottom ?

Moderate

A

$\sqrt{\frac{3}{4} gh}$
B

$\sqrt{\frac{4}{3} gh}$
C

$\sqrt{4 gh}$
D

$\sqrt{2 gh}$

Solution

Total K.E. of the cylinder at bottom is given by
$\begin{array}{l} K.E. = \frac{1}{2} {Iω}^{2} + \frac{1}{2} {Mv}^{2} \\ = \frac{1}{2} [\frac{{MR}^{2}}{2}] \frac{v^{2}}{R^{2}} + \frac{1}{2} {Mv}^{2} \\ = \frac{1}{2} [\frac{{Mv}^{2}}{2}] + \frac{1}{2} {Mv}^{2} = \frac{3}{4} {Mv}^{2} \end{array}$
Total K.E. at bottom
= loss of potential energy
$\begin{array}{ll} ∴ \frac{3}{4} {Mv}^{2} = Mgh \\ or v = \sqrt{[(4 / 3) gh]} \end{array}$

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