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

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

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