Question

A solid disc of radius of $a$ and mass $m$ rolls down without slipping on an inclined plane making an angle $\theta$ with the horizontal. The acceleration of the disc will be $\frac{2}{b}g\sin \theta$ where $b$ is(Round off to Nearest Integer) ($g=$ acceleration due to gravitiy; $\theta =$angle as shown in figure)

Answer 1

Since ball is rolling on the incline plane without slipping, the acceleration of the ball is given as: $$a=\frac{g\sin \theta }{1+\frac{k^2}{r^2}}$$ where $k$ is radius of gyration.

For disc, $\frac{k^2}{r^2}=\frac{1}{2}$

$\Rightarrow a=\frac{g\sin \theta }{3/2}=\frac{2}{3}g\sin \theta =\left(\frac{2}{b}g\sin \theta \right)$
$\Rightarrow b=3$