Question

A solid disc of radius '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 gravity, $\theta$ = angles as shown in figure)

(16th March 2nd Shift 2021)

Answer 1

3

Answer 2

For a solid disc rolling down an inclined plane without slipping, the linear acceleration $a$ is given by: $a = \frac{g \sin\theta}{1 + \frac{I}{ma^2}}$ where $I$ is the moment of inertia of the disc about its center. For a solid disc, $I = \frac{1}{2}ma^2$. Substituting the moment of inertia into the formula: $a = \frac{g \sin\theta}{1 + \frac{\frac{1}{2}ma^2}{ma^2}}$ $a = \frac{g \sin\theta}{1 + \frac{1}{2}}$ $a = \frac{g \sin\theta}{\frac{3}{2}}$ $a = \frac{2}{3}g \sin\theta$ The problem states the acceleration is $\frac{2}{b}g \sin\theta$. Comparing the two expressions for acceleration: $\frac{2}{b}g \sin\theta = \frac{2}{3}g \sin\theta$ This implies $\frac{2}{b} = \frac{2}{3}$, so $b = 3$. The question asks to round off to the nearest integer. Since $b=3$ is already an integer, the answer is 3.