How do you calculate kinetic energy in simple harmonic motion-pendulums?

The tricky part of this question is that it depends on where in the motion you are calculating the kinetic .

So the way a simple harmonic pendulum works is that a mass (m) is attached to a string with a certain length (L). The pendulum hangs at rest above the ground. However, as far as the pendulum is concerned, it is at its lowest possible point, so we will call that rest position, the ground level.

When the pendulum is pulled back, it is now a certain height (h) above its lowest or ground level. As it is held there, it has Potential Energy (stored energy or energy due to position). As it is held at rest at this height, the potential energy is also its total energy (TE). Potential energy (PE) is calculated as PE = mgh (mass x gravity x height). Therefore TE = PE.

When the mass is released, the pendulum accelerates as it loses height. So the PE decreases and velocity (v) increases. Now energy due to motion is known as Kinetic Energy (KE). Kinetic energy is calculated as KE = 1/2m##v^(2)##. So as height goes down, velocity goes up and PE is changed into KE. However, the total energy (TE) stays the same number. So when the pendulum reaches its lowest point (ground level) the TE=KE. But that KE is equal to the PE at the beginning.

For example if a 2.0 kg mass is pulled to a height to 1.8 m using an approximate gravity of 10 m/##s^(2)##, then TE = PE = ugh = (2.0)(10)(1.8) = 36 J of potential energy. When it reaches its lowest point, the TE = KE = 36 J. If you want to know its speed, then KE = 1/2m##v^(2)##, so 36 = 1/2(2.0)##v^(2)##, and v = 6.0 m/s.

If you want to know the KE at another height, like 1.0 m, well the TE is still 36 J. Now there is PE = (2.0)(10)(1.0) = 20 J, which means that the remaining 16 J is KE (KE = 16 J). You see, TE = PE + KE

If you want to see a more detailed lecture, check out and Energy