Determine the height h so that the particle with mass m just

Determine the height h so that the particle with mass m just has contact at the top of the loop with radius R in the cases: point particle (slides without friction) Ball with radius r and R = 5r (rolling without sliding)

Solution

to make the contact at the top of the loop, it should be

mv² / R = mg

The kinetic energy of the marble at the top consists out of rotational and translational energy

K = (Iw²)/2 + (mv² )/2

where we assumed that the particle is rolling over the track (no slipping). The moment of inertia of the particle is given by

I = 2mr² / 5

Using this expression we obtain for the kinetic energy

K = 7mv²/10

The particle will reach the top if

K = 7mv²/10 _> 7mgR /10

he total mechanical energy of the particle at the top of the loop-the-loop is equal to

E = K + U =  7mv²/10 + 2mgR

E = 7mgR /10 +  2mgR

E = 27 m g R / 10

The initial energy of the particle is just its potential energy at a height h

E = mgh

Conservation of energy now implies that

mgh = 27 m g R / 10

h = 27R / 10