Can someone help me with this physics problem? How do I even start?

Question

A block slides on a frictionless track starting at height H. It encounters a hill of radius "r". Develop an expression for the maximum value of H that *includes only g and r* that would allow the block to just barely remain in contact with the hill at its peak.

Answer 1

the key condition for remaining on the track is: gravity supplies the necessary centripetal acceleration required for circular motion.  (a = g = v^2/r)

the velocity acquired by a frictionless sliding object is, from conservation of energy:

Ep = Ek

mgH = (1/2)mv^2

thus, v^2 = 2gH

combining, we get:

g r = v^2 = 2 g H

H max = (1/2)r

A diagram would be helpful here.  In the above calculations, I'm taking H as "the distance above the top of the hill"  If your H is measured relative to the bottom of the hill, then:

H max = (3/2)r

Answer 2

I've seen a similar question to this posed with a loop rather than a hill, and it involves the initial (potential) energy of the block at height H (E = mgH), and the potential + kinetic energy at the top of the loop (E = mg(2r) + (1/2)mv²/2).

For the block to "just remain in contact" at the peak, the force of gravity (F = mg) must equal the centripetal force (F = mv²/r).

If this is the same problem, I recall that after all the substitutions and cancellations, the answer was H = 5r/2.

FWIW, my physics teacher even had a track with a ball bearing to illustrate it in class, and we verified it experimentally. ;)