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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 :

MechEngineer

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

[tex]U_{\theta} = mg\Delta h = mg(L - Lcos\theta) = mgL(1 - cos\theta)[/tex]

where g is the gravitational constant

The gravitational potential energy as a function of the angle, θ, measured from the vertical should be considered mgL(1 - cosθ).

Calculation of the  gravitational potential energy:

Since

At angle θ, the vertical distance from the mass m to the pivotal axle should be Lcosθ.

While at U=0, this distance is L.

So,

L - Lcosθ

Now

The change in potential energy should be

Uθ = mgΔh

= mg(L - Lcosθ)  

= mg(1 - cosθ)

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