A skateboarder is rolling back and forth inside a “frictionless half pipe” that is part of a circle of radius 10m. His motion is driven by gravity. If he starts at 2 m away from the bottom of the…

Question

A skateboarder is rolling back and forth inside a “frictionless half pipe” that is part of a circle of radius 10 m. His motion is driven by gravity. If he starts at 2 m away from the bottom of the pipe, how long will it take him to get back to where he started?

Answer 1

If you treat this as a small angle harmonic oscillator. Newton's 2nd law is mgsinθ = mRd²θ/dt² (F = ma) which is simplified to (θ~sinθ for small angles - here sin^-1(2/10) = 11 degrees) . This leads to a typical simple harmonic motion solution:

Rθ'' = gθ or θ'' = (g/R)θ ω = sqrt(g/R) and T = 1/(2πω) = 2πsqrt(R/g)

T is the period or the time to return to the initial position.