### The Gravitron

In the Gravitron, you enter a hollow cylinder, lean against the inside wall, and it starts to spin. When it's spinning sufficiently quickly, the floor drops away. You don't fall down because you are pinned to the wall. How does this work?

Use DID TASC.

Step 1: Draw a diagram and coordinate system. Step 2: Isolate system.

Choose +y = up and +x = toward the center.

Step 3: Draw all forces.

There are 3 forces acting. What are they?

Step 4: Take components. Step 5: Apply Newton's second law.

y-direction: Using ay=0 gives: fs = mg

x-direction: ΣFx= max = mv2/r   or   N = mv2/r= mω2r

Step 6: Solve.

To not fall, friction must balance the force of gravity. The maximum static friction force is: fs max = μs N = μs2r.

If the rotation is fast, fs max = μs2r > mg and there's nothing to worry about. As the speed decreases, the normal force and the consequent static friction force decrease. If friction drops below mg, a person on the wall would slide down.

The threshold for sliding occurs at an angular velocity:
ω2 =
 g μs r

Typical numbers:  r = 3, μ = 1, g = 10 leads to ω ≅ 1.8.