Question

Two round objects, of equal mass and equal radius released at top of an inclined plane rolls without slipping and reach bottom, one of them taking $25%$ more time, i.e. $\dfrac{{{t}_{1}}}{{{t}_{2}}}=\dfrac{4}{5}$ then:
A. Frictional force on slower one is more
B. Work done by friction on slower one is more
C. Both reach bottom with same kinetic energy
D. Radius of gyration of the faster body about the center of mass is more than that of others.

Answer 1

Hint: We know that $t=\sqrt{1+\dfrac{{{k}^{2}}}{{{R}^{2}}}}$ using this relation try to find out the relation between the frictional forces of the two bodies and also the relation between the radius of gyration (where $k$ is the radius of gyration). The formula for the frictional force is $f=\dfrac{mg\sin \theta }{1+\dfrac{{{k}^{2}}}{{{R}^{2}}}}$ using this relation try to find out the relation between the works done by the frictional force of both the bodies.

Complete step by step answer:
We know that the formula for the time is $t=\sqrt{1+\dfrac{{{k}^{2}}}{{{R}^{2}}}}$
Therefore the ratio of the times taken by both the bodies is
$\dfrac{{{t}_{1}}}{{{t}_{2}}}=\dfrac{\sqrt{1+\dfrac{k_{1}^{2}}{R_{1}^{2}}}}{\sqrt{1+\dfrac{k_{2}^{2}}{R_{2}^{2}}}}$
Given the ratio of times is $\dfrac{{{t}_{1}}}{{{t}_{2}}}=\dfrac{4}{5}$
It means that the second body is the slower one
And also given that both the bodies are having the same radius
Put these values in the above equation and solve for the relation between the radius of gyration of both the bodies.
After solving the above equation by using the given data we obtained the relation as
${{k}_{1}}<{{k}_{2}}$
Means the radius of gyration of the slower body is greater than the faster body
Hence option (D) is the incorrect option
 The formula for the frictional force is $f=\dfrac{mg\sin \theta }{1+\dfrac{{{k}^{2}}}{{{R}^{2}}}}$
Given the mass of both the bodies is same and using the given details we obtained the relation between the frictional forces as ${{f}_{1}}<{{f}_{_{2}}}$
This means the frictional force of the slower body is greater than the faster body

So, the correct answer is “Option A”.

Note: Radius of gyration or gyradius of a body about an axis of rotation is defined as the radial distance to a point which would have a moment of inertia the same as the body’s actual distribution of mass, if the total mass of the body were concentrated.