Physics rollercoaster question

Question

A rollercoaster is making a loop. At the top you are weightless and the coaster is moving at 20 m/s. What is the radius of the loop?

Answer 1

At the top of the loop you would have the possibility of two forces acting on you, both would be directed downwards. The first is gravity, the second is a normal force. If you are "weightless," gravity is still present, but you are not experiencing a normal force. That is the only force, of the two, that you can "feel." Since you are moving in a circle, Newton's 2nd Law states:

F_Net = m (v²)/r

Since the only force present is gravity, that becomes:

F_G = mg = m(v²)/r

Rearranging to solve for r gives:

r = (v²)/g = (20)²/9.8 (or 10 depending on which value of g you use in your course)

r = 40.8 m (or 40 m if g is 10)

Answer 2

This is a case in vertical circular motion where the Normal Force of the track has a value of zero. If you are traveling any faster you lose contact with the track.

if you draw the free body diagram you will see that the only force is mg down, therefore the resulting net force can be written

mv²/r =mg

The mass m cancels so we get

r=v²/g

Using g as approx 10, we get the radius of 40 meters