An Example Question-
Three bodies, a ring, a solid cylinder and a solid sphere roll down the same inclined plane without slipping. They start from rest. The radii of the bodies are identical. Which of the bodies reaches the ground with maximum velocity?
We can easily derive the expression for maximum velocity attained – $${\rm{v = }}\sqrt {{{{\rm{2gh}}} \over {\left( {{\rm{1 + }}{{{{\rm{k}}^{\rm{2}}}} \over {{{\rm{R}}^{\rm{2}}}}}} \right)}}} $$So the object that has the least k2/R2 value attains the maximum velocity.
The answer is Solid Sphere.
We expect Moment of Inertia to be the factor that decides the velocity attained by an object in rolling motion. But two objects can have the same moment of inertia and still need not attain the same velocity after rolling certain distance. Its all dependent on the k2/R2 value of the particular object.
What does k2/R2 (k – Radius of Gyration and R – Radius of The Object) mean in real world and how to think about it intuitively?
Among Two Bodies having same $Radius\;of\;Gyration$, the one having greater radius attains the maximum velocity. Why is that?
Best Answer
The radius of gyration tells you something about the distribution of mass about an axis of rotation.
The "further" the mass is from the axis of rotation the larger is the radius of gyration.
As a body rolls down a slope without slipping it gains both translational kinetic energy $\frac 12 mv^2_{\rm com}$ and rotational kinetic energy $\frac 12 I\omega^2$ with the constraint that $v_{\rm com}=R\omega$.
The larger $k$ and hence the moment of inertia $mk^2$.
This would mean a greater proportion of the available energy (from the loss of gravitational potential energy decrease) would go into increasing the rotational kinetic energy rather than the translational kinetic energy.
For the ring $k=R$ as all the mass is distributed at a distance $R$ from the axis.
The cylinder of radius $R$ obviously has some mass closer to the axis than the ring so its radius of gyration will be less.
Then for a sphere there is even more mass nearer to the axis of rotation.
When a body is rotating the further the mass is away from the axis of rotation the faster it is moving and so its contribution to the rotational kinetic energy is greater.
The ratio $\frac kR$ is the factor which shows how the mass is distributed relative to the radius of the body.
If $k$ is kept constant and $R$ is increased this implies that more mass is concentrated nearer the axis of rotation.
Thus less rotational kinetic energy is gained compared to the gain in translational kinetic energy - the translational speed of the centre of mass of the body will be larger.