Potential Energy of a Pendulum

[velo city]

What is the potential energy of a pendulum at any point?

 

[S. Moger]

Assuming a standard pendulum?

The potential energy can be computed through m g h. So you need to find an expression for h at "any point", which you can do through knowing the radius of the trajectory and the angle.

 

[BrettJimison]

Welllllll...
If you say the pendulums length is "L" then when the pendulum swings "up" through an angle θ it will be displaced a vertical displacement "h".
Make a right triangle with θ at the hinge point. The hypotenuse is "L" and the adjacent side is "x"
so cosθ=x/L so x=Lcosθ.
But if x is the adjacent side, it is also equal to L-h, sooo..
L-h=Lcosθ which means h=L-Lcosθ.
Thus Potential energy at any point is P.E=mg(L-Lcosθ)

 

[BrettJimison]

Note:
I am assuming the pendulum "string" has a negligible mass as well..

 

[SpY]]

Hi, I just have a related question on the energies of a pendulum, and am not looking to start a new thread (unless mod feels it should be).

F the potential energy and kinetic energy are given by
U(θ)=mgLθ )
K(θ)= mgL(cosθ_max-cosθ)
*Since kinetic energy simply is difference between mechanical and potential energy, where the total mechanical energy is equal to the potential energy at θ_max

I'm trying to plot or find the equation of a graph for Energy vs. Time. It's obvious that energy vs. theta will be sinusoidal, however if theta is itself sinusoidal as a function of time (SHM), what would the graph of energy vs time look like? I'm struggling to convince myself it will also be sinusoidal.

I'd also like to consider how the graph would be different for small angles (sinθ~θ) and for θ~90° (unless it involves analysing elliptic integrals)

Anyone can point me in a direction to start?