Question

Which of the following order is true for the time taken by the bodies to reach the bottom of incline?
\[\begin{align}
  & A)sphere>disc>ring \\
 & B)ring>disc>sphere \\
 & C)disc>ring>sphere \\
 & D)disc=ring=sphere \\
\end{align}\]

Answer 1

Hint : Assume the ring, sphere and disc of mass\[M\], and each body has some friction working against them. But keep in mind the motion is pure rolling, and hence the point of contact of the mass does not change. Also, keep in mind that the angle of incline is the same in all three cases.

Complete step-by-step solution:

We assume that all three bodies have the same mass. But due to different moments of inertia, the amount of friction applied on each body is also different.
Let us calculate the forces applied on these bodies. We know that gravity is acting on the three bodies but there is no force other than gravitational force because pure rolling motion is in effect.
In pure rolling motion, one point to be noted is that the point of contact between the slope and the object remains the same, and hence the frictional force doesn’t work here and is counted as zero.
Let \[{W_f}\]is the work done due to frictional force.
So, \[{W_f}\]= 0
Let \[{W_g}\] be the work due to gravitational force.
And now only the other force is gravity is acting, and then the work is done by gravity only.
And since, the height of incline is the same for all three bodies.
So,\[{W_g}\] is also same for all three bodies
Now, \[\Delta P.E.\]represents change in potential energy which is also the same for all three bodies since the mass, gravity and height is the same.
So, using conservation of energy theorem,
\[\Delta K.E.\]Represents change in kinetic energy which is also the same for all three bodies.
Now, if Kinetic energy is same for all three bodies, and mass is also same, that means the velocity of the three bodies will also be the same.
And if velocity is the same, that means the time at which the bodies come to the bottom of the incline will also be the same.
So, it is concluded that the correct answer will be option (D).

Note: When bodies are moving with pure rolling motion, it is to be noted that pure rolling motion only comes into effect when there is no resistance. Some friction is necessary to get the body to roll, but once it starts pure rolling it doesn’t need more friction. In pure rolling, the body starts rolling without slipping, while in impure rolling the body will slip while rolling.