Conservation of Energy and the Race between Different Shapes on an Inclined Ramp

[robertmatthew]

Homework Statement

Three uniform objects: a ring, a disk, and a sphere all have identical masses and radii. They are released simultaneously from the top of a ramp 3.80 meters in LENGTH. If the ramp is inclined at θ= 17.0°, use conservation of energy to calculate the linear speed of the ring, disk and sphere.

Homework Equations

Ei=Ef
I_ring = mr²
I_disk = (mr²)/2
I_sphere = (2mr²)/5

The Attempt at a Solution

Ui + Ki = Uf + Kf
mgh = (1/2)Iω²_f + (1/)mv²_f <--initial K and final U are zero

then substituting I for I_ring
mgh = (1/2)((mr²)(v_f)/(r)) + (mv²_f)
masses cancel, pulled out 1/2
gh = (1/2)(rv_f + v²_f)

But that's as far as I can get before I can't understand it.

 

[TSny]

The Attempt at a Solution

Ui + Ki = Uf + Kf
mgh = (1/2)Iω²_f + (1/)mv²_f <--initial K and final U are zero

then substituting I for I_ring
mgh = (1/2)((mr²)(v_f)/(r)) + (mv²_f)

Note that ω_f is squared.

 

[robertmatthew]

I always end up asking questions on here because of stupid mistakes like that, haha. That made much more sense, with the radii canceling out. Thanks so much.