Simple energy question with rotation

[oreosama]

Homework Statement

a small solid sphere of mass m and radius r starts from rest and rolls down a hill without slipping. the sphere encounters a loop of radius R where R >> r.

Given R determine the min height h such that the ball remains on the track throughout the loop

Homework Equations

I = 2/5 mR^2

ω = v/r

The Attempt at a Solution

highest point requiring most energy is at top of loop:

mgh = mg2R + 1/2 Iω^2

mgh = mg2R + 1/2 * 2/5 m r^2 *v^2/r^2

gh = 2Rg + 1/5*v^2



at top of loop, normal force is 0,

Fy = N + mg
ma = mg

m*v^2/R = mg
v^2 = gR


...

gh = 2Rg + 1/5 Rg

h = 11/5 R

this is correct?

 

[Simon Bridge]

To check your results - it is useful to compare to something:

i.e. If this were a block sliding around the track without friction, then what would the minimum h have to be?
(for the ball, some energy is stored in rotation - so you need to start higher than this.)

How fast is the ball rolling at the top of the hoop?
(i.e. is the kinetic energy a lot or a little more than the potential energy there? says if you should start a little higher or a lot higher.)

 

[TSny]

mgh = mg2R + 1/2 Iω^2

Have you included all of the kinetic energy at the top?