How to Determine Velocities for Rolling Objects on an Inclined Plane?

[Granger]

Homework Statement

Side by side on the top of an incline plan with height=2 meters a cylinder (Ic= MR^2/2) and a sphere (Ie=2MR^2/5) with equal radius, that come down to the base, rolling without slipping. Mass of the cylinder = 2.0 kg; Mass of a sphere=4.0 kg.

Homework Equations

$$K_r= 1/2 I \omega ^2$$
For rolling without slipping $$v=\omega R$$

The Attempt at a Solution

At first I thought this was a pretty linear problem.
Applying both equations to both the sphere and the cylinder:

$$K= 1/4 M_c v_c^2$$
$$K= 1/5 M_e v_e^2$$

Than I applied conservation of mechanical energy to determine velocity. However this is not correct since we don't know if there is a friction force (in fact we find out in the next question that it has).
So how should I proceed in this case to determine the velocities?
Thanks!

 

[Orodruin]

You forgot about the energy due to the motion of the centre of mass.

The frictional force does no work since the objects are rolling without slipping.

Edit: You have also not stated what the actual question is ...

 

[Granger]

I realized I was understanding the concept wrongly. Even though the ball is not slipping the center of mass moves so it has a kinetic energy associated, right?
It has nothing to do with the ball slipping.
Thanks!

 

[Orodruin]

I realized I was understanding the concept wrongly. Even though the ball is not slipping the center of mass moves so it has a kinetic energy associated, right?

Yes. In general the total kinetic energy can be written as the energy related to the motion of the centre of mass and an additional piece due to the rotation (using the moment of inertia relative to the centre of mass).

It has nothing to do with the ball slipping.

If the ball was slipping you would have to worry about frictional forces (unless specified that friction can be neglected). In order for the ball to roll, there needs to be friction, but it will not perform any work if the ball rolls without slipping.