You are riding a rollercoaster going around a vertical loop, on the inside of the loop. If the loop has a radius of 30 meters, how fast must the cart be moving in order for you to feel three times as heavy at the top of the loop

v = 24.2m/s

Explanation:

Given R = 30m

The two forces acting on you while on the roller coaster ride are the normal force and your weight.

By Newton's second law

N - W = mv²/R

For you to feel 3 time as heavy, the normal force of the seat of the roller coaster on you must be 3 times your weight. That is

N = 3*W = 3*mg

W = mg

3mg - mg = mv²/R

2mg = mv²/R

2g = v²/R

v² = 2gR

v = √2gR

v = √ (2*9.8*30)

v = 24.2m/s

Speed of the cart at the top of the loop = 34.3 m/s

Explanation:

Gravitational acceleration = g = 9.81 m/s2

Your mass = m

You feel three times as heavy at the top of the loop.

Normal force on you = N = 3mg

Radius of the loop = R

Speed of the cart = V

Centripetal force required for the circular motion = Fc

F = m

The centripetal force is provided by the normal force on you which is directed downwards and your own weight which is directed downwards.

Fc = mg + N

Fc = mg + 3mg

Fc = 4mg

m12 - = 4mg R

V = 4gR

V = 4 (9.81) (30)

V = 34.3 m/s

Speed of the cart at the top of the loop = 34.3 m/s