Question

A solid sphere rolls down without slipping from rest on a $30^\circ$ include. Its linear acceleration is
$\left( A \right)\dfrac{{5g}}{7}$
$\left( B \right)\dfrac{{5g}}{{14}}$
$\left( C \right)\dfrac{{2g}}{3}$
$\left( D \right)\dfrac{g}{3}$

Answer 1

First of all in this problem we have to make a rough diagram to solve this by representing all the parameters. Now using the acceleration formula for the rolling object as shown by the figure. The acceleration gives us the formula that has a relationship between the angles, radius of gyration, acceleration, and acceleration due to gravity. On putting the values we will find our solution.

Complete step by step answer:
As per the problem we have a solid sphere that rolls down without slipping from rest on a $30^\circ$ include.
Now we need to calculate the linear acceleration.

We know that linear acceleration of a rolling sphere in incline plane will be given as,
$a = \dfrac{{g\sin \theta }}{{1 + \dfrac{{{K^2}}}{{{R^2}}}}} \ldots \ldots \left( 1 \right)$
Where,
$g$ is equal to acceleration due to gravity in meters per second.
$\theta$ is the angle incline.
$K$ is the radius of gyration.
$R$ is the radius of the sphere.
$a$ is the linear acceleration.
We know the moment of inertia of a sphere is represented as,
$\dfrac{2}{5}M{R^2} = M{K^2}$
Cancelling the common terms we will get,
$\dfrac{2}{5}{R^2} = {K^2}$
$\theta = 30^\circ$
Now putting the known values in the equation $\left( 1 \right)$ we will get,
$a = \dfrac{{g\sin 30^\circ }}{{1 + \dfrac{{\dfrac{2}{5}{R^2}}}{{{R^2}}}}}$
Cancelling the common term we will get,
$a = \dfrac{{g\sin 30^\circ }}{{1 + \dfrac{2}{5}}}$
On further solving we will get,
$a = \dfrac{{g\dfrac{1}{2}}}{{\dfrac{7}{5}}}$
$\Rightarrow a = \dfrac{{5g}}{{14}}$
Therefore the correct option is $\left( B \right)$.

Note: Remember that the radius of gyration is defined as the radial distance to a point that would have a moment of inertia the same as that of the actual distribution of mass if the total mass of the body were concentrated there about the axis of rotation.