Loop de loop question

Question

I have an assignment in physics and I can't understand the difference between the 2 questions. That's the question

A skier with mass m=70 kg stands on a ramp of height h. At the foot of the ramp, there is a loop with a radius of r=15 m. When entering the ramp, the speed is v1=0 m/s.

a) For test purposes, the skier is also held in the loop by a rail system so that he cannot fall off if his speed is too low. How high does the ramp have to be at least so that he can get through the loop in this case?

b) Normal operation of the loop takes place without a safety rail system. In this case, how high does the ramp have to be so that the skier can get through the loop without falling off? Here I thought that the force of gravity must be equal to the centripetal force.

I can't understand what difference that makes if the person is held by a rail system or not. I tried to solve it in 2 ways considering the potential and kinetic energy, so I'm not sure that my answer is correct. It would be really helpful is someone could say that what I thought makes sense.

Thank you

Mgh=0.5mv*2+mg(2R)

Mgh=0.5mgr+mg(2R)

h=0.5R+2R

h=2.5R

h= 15m x 2.5=37.5m

I got this answer using conservation of energy but I still cant understand to which part of this question this can be referred to and why.

I would appreciate any help

Answer 1

With the safety rail you only need enough energy to make it to the top of the loop.

Without the safety rail you need enough energy to counteract the gravitational energy at the top of the loop.

The answer you gave is the correct answer for part b, because you need the centripetal force to be strong enough to counteract gravity. This can nicely be worked out in terms of conservation of energy as you have done. Note that the left hand side of your equation is the total initial energy and the right hand side is the total energy at the top of the loop.

Part a is actually even simpler because in this case you don't have to worry at all about the centripetal force, you just need to convert enough potential energy to kinetic energy in order to balance the potential energy at the top of the loop.