rotational motion?

Question

A solid sphere, a hollow sphere and a disc, all having same mass and radius, are placed at the top of a smooth incline and released. How do we prove that all will take same time in reaching the bottom

Answer 1

The only way these three objects have the same mass and diameter, is that they are made of different materials. These materials must have exact same physical properties (surface hardness, rigidity, elasticity, thermal properties, mass density and distribution, etc) for this to be a fair comparison.

Assuming they have identical properties, this experiment is carried out in normal atmosphere, and the ramp is long enough, they will NOT reach the bottom at the same time, as air resistance will play an important role and the terminal velocities will not be the same. The disk faces least air resistance due to its shape and will therefore reach a higher speed.

One addition/correction to the answer minootoo gave you:
"When you drop feather and a ball from a height it reaches the ground at same time. ..."

I am very surprised by the statement made.
The ball and feather dropped from the same height will only reach the ground together in the VACUUM.

In normal atmosphere, the air resistance has much greater effect on the feather. This is true even if masses of these objects were the same.

Answer 2

If they're rolling they take different times because not only does the mass get accelerated but so does the moment of inertia (i.e., "rotational motion"), and the ratio of moment of inertia to mass is different for each shape. The ref. gives the details for a rolling solid ball. If they are sliding without rolling (which "smooth" implies) then they reach the bottom simultaneously.
EDIT: The 2nd answer has it right except the acceleration is g*sin(theta)/(1+γ); the mass has to accelerate as well as the moment of inertia. Thus a rolling solid sphere accelerates at 5/7 (71%) the rate of the same sphere sliding, a disk at 2/3 (67%), and a hollow sphere at 3/5 (60%).

Answer 3

Does "smooth incline" mean that they SLIDE (without rolling)? Or do they ROLL?

It makes a big difference; because if they roll, then they DON'T all take the same time!

The time it takes for an object to ROLL down a ramp, depends on the ratio of its moment of inertia to its mass, which in turn depends on how its mass is distributed.

Specifically, a rolling object's (linear) acceleration down an incline of angle θ is given by:

a = g(sinθ) / γ

where γ is a constant that depends on the mass distribution of the rolling object.

For a solid sphere, γ = 2/5
For a hollow sphere, γ = 2/3
For a disk, γ = 1/2

In this case, the solid sphere will reach the bottom first; then the disk, then the hollow sphere.

Answer 4

When you drop feather and a ball from a height it reaches the ground at same time.

Because this motion is independent of mass.

But In case of the spheres even though it is rolling down an incline and granted that such three different spheres are possible and the friction is the same, then that reduces to falling body. If you examine the derived Equation of this motion you will find it to be independent of mass.

That is how you prove it.

Answer 5

give 5 examples in which both motions are combine rotatory and translatory