# Roller Coaster Loops

Updated: Jan 9, 2020 As with any problem, we must decide what lens/perspective we will look at the problem with. Will it be energy or kinematics? In this case both perspectives will work, although energy is more intuitive.

### Energy

In this problem, energy is conserved as gravity is a conservative force. Normal force does no work as it will be acting perpendicular to the velocity, possibly changing its direction BUT NOT magnitude. (Notice that mass does not matter in this case. If a physics problem appears to give you too little information, try to solve it algebraically and see if any variables cancel out)

Setting the gravitational potential energy = kinetic energy, we arrive at the final equation. However, we don’t know what the velocity is. How do we solve for it?

We must look at some concepts relating specifically to circular motion. Whenever you encounter a problem with riders inside a roller coaster loop, it will deal with either the top of the loop, the bottom of the loop, or both.

Analyzing the force diagrams of both positions we arrive at: Notice that at the bottom of the loop the forces act in opposite directions and at the top of the loop, the forces act in the same direction. This will affect how net force is calculated.

At the bottom: Fnet = Fnorm – Fgrav

At the top: Fnet = Fnorm + Fgrav

Fnet will always point towards the center of the loop due to circular motion.

For this particular problem, we will be focusing on the top of the loop instead of the bottom as we are aiming to “complete the… loop”. Reaching the bottom should be no problem, but reaching the top takes some calculation.

To relate the net force to velocity, we must also know that:

Centripetal Acceleration = V^2/R (where V is the velocity and R is the radius of the circle)

Using F=MA, we arrive at: