# What is the total kinetic energy of a pendulum?

I have a sphere of a mass $$m$$ and radius $$r$$, attached to a massless string with a length $$l$$. The pendulum moves with an angular velocity $$\omega$$.

I know that axes of rotation can be relative. If I chose that one going through a pivot, then I would write the toal kinetic energy as

$$E_k=\frac 12[\frac 25 mr^2+m(r+l)^2]{\omega}^2 .$$

But I saw another way to calculate it: the axis went through the centre of mass of the sphere and I somehow demeed the movement translational. Then the centre of mass had a velocity $$v$$ and the energy was then

$$E_k=\frac 12 \frac 25 mr^2 {\omega}^2+\frac 12 mv^2$$,

which is the same since $$v=\omega (r+l)$$.

But I do not undertstand how is it possible to have such axis of rotation? How can it go through the centre of mass? It does not rotate around it. Or yes from a specific point of view?

And how can I consider the motion to be translational when all point making the sphere do not have the same trajectory?

• Presumably you mean maximum kinetic energy rather than total KE, since the KE oscillates between zero at the extremes of each swing and some maximum value at the centre point. Commented Jul 7, 2023 at 17:44
• When I think about, why is the energy in my first equation constant? The only possible way it can change is by changing the angular velocity, but that remains constant, doesn't it? So how is that possible when the kinetic energy is zero at the maximum point? Commented Jul 7, 2023 at 18:14
• $\omega$ is the angular velocity of the pendulum bob, which clearly changes over the course of a swing. The frequency of the pendulum's oscillation is constant (assuming there is no friction or drag), but that is a different quantity. Commented Jul 8, 2023 at 8:05