Calculating G-Forces in a Banked Turn on a Roller Coaster

[ntweisen]

Hello. I’m trying to calculate the horizontal and vertical gs felt by passengers on a roller coaster going through a banked turn. This is for the ideal banked curve of angle theta where no friction is required to keep the car from sliding to the outside or inside of the curve.

Homework Equations

F = m *( v^2 )/ r

The Attempt at a Solution

I came up with this formula but I don't think it is correct: G’s felt = 1/sin(theta)

 

[CWatters]

Assuming the coaster is on a planet like the Earth then there are two forces acting on the passengers...

1)The centripetal force due to the curved track and
2)That due to gravity.

These two forces are vectors that are probably pointing in different directions, so you need to do vector addition to work out the resultant force. Then convert to the g equivalent.

If they happen to point in the same direction (eg at the top or bottom of a vertical loop) then it's a bit easier to add them.

 

[CWatters]

PS. The banking has no effect on the magnitude of the centripetal force or it's direction relative to gravity. Although obviously it does change how the person perceves it. What matters is the plane of rotation and it's angle relative to gravity.

Compare an "inside loop" (feels like +ve g) with an "outside loop" (feels like -ve g) although the magnitude of the force would be the same for a given radius and speed etc.

PPS When you do the vector addition it might make things easier if you use the method that converts each vector to x and y components first before adding them. Saves a step later.