Question

Three bodies a ring a solid cylinder and a solid souere roll down the same inclined plane without slipping at same time which would have maximum velocity

Answer 1

We know that energy of a rolling body is conserved. So,

Potential energy lost by the body = kinetic energy gained

As the body starts from rest, the kinetic energy gained will be equal to the final kinetic energy of the body.

Answer 2

If these objects slid down a frictionless ramp from the same height, they would reach the bottom in equal time. The masses nor their densities matter. The potential energy is M*g*h near the surface of the earth. (Where M=mass of the object, g=the gravitational constant [9.8 m/s^2] and h=height fallen.) The kinetic energy of these objects is 1/2 *M*v^2. (Where v is the velocity at the bottom.) Setting these two equal, M*g*h=1/2 *M*v^2. Solving for v^2=2*g*h. The final velocity, and thus travel time does notdepend on mass.

Rolling changes things. Now, part of the potential energy turns into rotational kinetic energy, which I will simply call rotational energy. The object which contains the most rotational energy, has the least translational kinetic energy, travels slowest, and reaches the bottom last. The rotational kinetic energy is 1/2 *c*M*v^2, where c is the coefficient governing the type of rolling object. Now the potential energy is distributed between both rotational and kinetic energies according to the equation M*g*h=1/2 *M*(c+1)*v^2. Again, mass is irrelevant, and v^2=2*g*h/(c+1). The larger the value of c, the slower the final velocity and the longer it takes for the object to reach the bottom.

The hollow cylinder has a value of c=1. This is the largest value c can be, because the rotating mass is farthest from the center of the object. In this case, the most energy possible becomes rotational energy, so the object moves slowest and hits bottom last.

The solid cylinder has a value of c=1/2. This is sensible because some mass is in the center axis and some is on the edge. If the mass is all central, c=0. So, this averages to c=1/2. On the other hand, a solid sphere has a value of c=2/5. This value is less than that of the cylinder because only a small portion of the sphere is the full radial distance from the central axis of rotation. As we approach the axis of rotation, more of the sphere’s mass has less and less of the rotational energy. So, with a c=2/5, the sphere (at the bottom) has the least rotational energy, has the largest kinetic energy, and reaches the bottom first.