A Simple Pendulum and Friction problem

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The discussion centers on calculating the work done by the frictional force on a simple pendulum after one complete period. The equation provided, ((beta)^2)'= (2g/l)(cosbeta - costheta), is key to understanding the motion, where theta represents the maximum angle of the pendulum's swing, and l is the length of the pendulum string. The user initially seeks guidance on how to apply this equation to determine the path taken by the pendulum for integrating the force of friction. However, the user later concludes that they resolved the question independently, deeming it straightforward and apologizing for any perceived inconvenience caused by their request for help.
Ed Quanta
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Im supposed to calculate the work done by the frictional force on a simple pendulum after one period. I have to use the following equation to do this calculation.

((beta)^2)'= (2g/l)(cosbeta - costheta) where thetha is the the maximum angle of the motion.
In other words theta is the angle at which the derivative of beta=0. l is the length of the pendulum string.

How do I use this equation to determine the path taken by the pendulum so I know what I am integrating the force of friction over?

By the way, I am given information regarding the initial position of the ball above the floor, the length of the pendulum wire, and the ceiling height.
 
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Nevermind the help, I figured it out. It was an easy question. Sorry for wasting internet space.
 

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For simple comparison, I think the same thought process can be followed as a block slides down a hill, - for block down hill, simple starting PE of mgh to final max KE 0.5mv^2 - comparing PE1 to max KE2 would result in finding the work friction did through the process. efficiency is just 100*KE2/PE1. If a mousetrap car travels along a flat surface, a starting PE of 0.5 k th^2 can be measured and maximum velocity of the car can also be measured. If energy efficiency is defined by...
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