Physics

When the pendulum just barely completes a circular motion around its pivot, the tension in the string at the topmost point of the circular path is zero. At the top of the circle, the only force acting on the pendulum bob is its weight .

 

The centripetal force required to complete the circular motion at the top is provided by the weight of the bob. The centripetal force , where is the speed of the bob at the top of the circle and is the length of the pendulum.

 

At the top of the circle, , so , which gives

.

 

Use the conservation of mechanical energy to find the energy at the bottom:

The total mechanical energy of the pendulum is conserved. Let's consider the potential energy at the top of the circle and the kinetic energy at the bottom of the circle.

 

The potential energy at the top of the circle with respect to the bottom of the circle is

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