Speed of a Pendulum

A simple pendulum consists of a ball at the end of a light string. If the length of the string is 1.4 m and the ball is released from rest at a point where the string makes an angle of 20 degrees with the vertical, what is the maximum speed of the pendulum?
















Use the master energy equation:

Ui + Ki + Wnc = Uf + Kf

i represents the initial position, where the mass is released from. f is the final position, at the bottom of the path. Set the zero level for potential energy to be the bottom of the path.

The initial kinetic energy Ki = 0 because the ball is released from rest
The work done by non-conservative forces Wnc = 0 because we neglect air resistance.
The final potential energy Uf = 0 because we set the zero for potential energy to be the bottom of the path. So:

Ui = Kf

mgh = ½ mvf2, where h is the height from which the ball is released.

The m's cancel, leaving:

vf = (2gh)½

h can be determined from geometry. If L is the length of the string:

h = L - Lcos(θ) = L [1-cos(θ)]

With L = 1.40 m and θ = 20 degrees, h = 0.0844 m.

This gives vf = (2gh)½ = 1.29 m/s