Different round objects of same mass on an incline

Homework Statement

A solid sphere, a hollow sphere and a disc, all having the same mass and radius are placed at the top on an incline and released. The friction coefficients between objects and the incline are same and not sufficient to allow pure rolling. Least time will be taken in reaching bottom by? Which object will have the least kinetic energy on reaching the bottom?

The Attempt at a Solution

Since the friction is same for all, their accelerations down the plane will be same. Hence time taken will be same for all.
This implies that velocities at the bottom should be same (since they start from rest and reach in same time). Therefore the Kinetic energies must be equal.

But the answer is 'the hollow sphere' (for the second part)

One more thing, suppose the objects are in pure rolling motion (friction is sufficient). Then in that case will the time taken to reach the bottom of incline be same for all?

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lewando
Homework Helper
Gold Member

Since the friction is same for all, their accelerations down the plane will be same. Hence time taken will be same for all.
If there were no friction, this would be true. These things have to get rolling before they can come down the incline.

Doc Al
Mentor

Since the friction is same for all, their accelerations down the plane will be same. Hence time taken will be same for all.
Good.
This implies that velocities at the bottom should be same (since they start from rest and reach in same time). Therefore the Kinetic energies must be equal.
Their translational velocities will be the same. But what about their rotational velocities?

One more thing, suppose the objects are in pure rolling motion (friction is sufficient). Then in that case will the time taken to reach the bottom of incline be same for all?
What do you base this on? Hint: Don't assume that the static friction is the same for all.

Their translational velocities will be the same. But what about their rotational velocities?
Loss of potential energy of each object appears as gain in Kinetic (translational and rotational) energy. There is no energy loss anywhere.
So the kinetic energies gained by them should be equal - right?
rotational velocity will be maximum for the solid sphere and minimum for the hollow sphere.

What do you base this on? Hint: Don't assume that the static friction is the same for all.
base on what? did not get you.

lewando
Homework Helper
Gold Member

Consider the moments of inertia of the 3 objects, this may help. Which is the hardest to get spinning?

Solid sphere is hardest to get spinning. What next?

lewando
Homework Helper
Gold Member

Are you sure about that? I thought that the moment of inertial had to do with mass distribution about a radius and also there is an r-squared effect too (mass at a greater value of r contributes to MoI more than mass near the center). Based on this, the shell would be my bet.

Yes you are right, moment of inertia of shell is greatest.
But L=Iw
since L is constant for all bodies, w is inversely proportional to I
so sphere is hardest to spin

Doc Al
Mentor

Loss of potential energy of each object appears as gain in Kinetic (translational and rotational) energy. There is no energy loss anywhere.
So the kinetic energies gained by them should be equal - right?
rotational velocity will be maximum for the solid sphere and minimum for the hollow sphere.
If there is slipping, there will be energy loss.

base on what? did not get you.
I want your reasons for thinking that the time would be the same if they rolled without slipping.

I want your reasons for thinking that the time would be the same if they rolled without slipping.
I got it. Time taken will be different because friction force acting would be different (as u told) and hence different acceleration. Thanks

lewando
Homework Helper
Gold Member

But L=Iw
since L is constant for all bodies, w is inversely proportional to I
so sphere is hardest to spin
Not sure I agree that L is constant for all bodies (please don't make be actually do this problem analytically).

I do agree that solid sphere has the smallest I.

Take this to the extreme: A central point mass on a massless disk is not going to waste much time trying to rotate. It will get to the bottom as fast as a point mass sliding down a frictionless incline.