Question

A pendulum consists of a massless stiff rod of length L hanging from a nearly frictionless axle, with a mass m at the end of the rod. Calculate the gravitational potential energy as a function of the angle, θ, measured from the vertical. Set U = 0 at the location of the mass when the pendulum is hanging straight down.

Answer 1

Answer:

mgL(1 - cosθ)

Explanation:

At angle θ, the vertical distance from the mass m to the pivotal axle is Lcosθ. While at U=0, this distance is L. Due to the mass is hanging straight down. Therefore the vertical distance from this lowest point to point at angle θ is

L - Lcosθ

The change in potential energy would be this height difference times mg

$U_{\theta} = mg\Delta h = mg(L - Lcos\theta) = mgL(1 - cos\theta)$

where g is the gravitational constant