First slide

Pure rotational motion and its kinetic energy

Question

A cylinder of mass 4 m and a block of mass m is placed on an inclined plane as shown in the figure. The linear acceleration of cylinder is

Difficult

A

$2 m / s^{2}$
B

$4 m / s^{2}$
C

$\frac{10}{7} m / s^{2}$
D

$\frac{3}{7} m / s^{2}$

Solution

The velocity of the block and the velocity of the top point of the cylinder same i.e., doble the velocity of center of mass, from conservation of energy.
$\begin{array}{l} m g h = \frac{1}{2} m υ^{2} + \frac{1}{2} 4 m {(\frac{υ}{2})}^{2} (1 + \frac{k^{2}}{R^{2}}) \\ m g h = \frac{1}{2} m υ^{2} + \frac{1}{2} 4 m {(\frac{υ}{2})}^{2} (1 + \frac{1}{2}) \\ = \frac{1}{2} m υ^{2} + \frac{1}{2} m υ^{2} (\frac{3}{2}) \\ = \frac{1}{2} m υ^{2} (1 + \frac{3}{2}) \\ m g h = \frac{1}{2} m υ^{2} (\frac{5}{2}) \\ \frac{4 g h}{5} = υ^{2} υ^{2} = 2 a h \\ 2 a = \frac{4 g}{5} \\ a = \frac{2 g}{5} = 4 m / s^{2} \end{array}$

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