How Does Rotational Inertia Affect the Motion of a Sphere on a Moving Ramp?

[Benton]

1. Homework Statement
A solid sphere (mass of m, radius of r and I=2/5 mr²) is rolling without slipping on
a rough surface with a speed of v. A ramp (mass of 2m and angle of θ) rests on a
smooth surface and is free to slide on the surface. As the ball rolls up the ramp, the
ramp begins to move. Provide all answers in terms of the given variables and any
fundamental constants.

A. What will be the speed of the ball/ramp when the ball reaches its highest point on
the ramp?

B. What distance L will the ball roll up the incline?

C. What will be the speeds of the ball and the ramp after the ball rolls back down off
of the ramp?

Homework Equations

Conservation of Momentum, Conservation of Energy

The Attempt at a Solution

A. mv=(m+2m)v
mv=3mv
v_f=1/3v

B.
we now have v_f which is v/3 we can solve

ΣE_i=1/2 Iω²+1/2mv²=

1/2(2/5mr²)(v/r)²+1/2mv²=

1/5mr²+1/2mv²=7/10mv²

ΣE_f=mgh+1/2*3mv_f²=

mgh+1/2*3m(v/3)²=

mgh+1/6mv²=7/10 mv²

h=8v²/15g

L=(8v²/15g) cscΘ

c. I would like guidance on how to solve c.
I know it uses energy, but I'm stuck on this one

 

[Doc Al]

c. I would like guidance on how to solve c.
I know it uses energy, but I'm stuck on this one

Once again, momentum and energy are conserved.

 

[mfb]

You'll need both energy and momentum for (c). It is probably easier to start in the frame where (ramp+ball) start at rest.

Technical detail for (b): It is beyond the scope of this problem, but in general you will get losses due to internal friction when a ball hits an incline like that.