Why doesn't tension do any work in a vertical circular motion?

Question

Suppose a ball is undergoing vertical circular motion while being tied to a fixed point with a string. Denote the angle of the string with the vertical as $theta$ and string length as $l$. Given the speed at the highest point be $v_0$, then the speed $v$ at angle $theta$ can be found by solving $frac{1}{2} mv_0^2 + mgl= frac{1}{2}mv^2 + mglcostheta$. The tension should be a variable in this kind of circular motion, but why is it missing from the equation (or why doesn't it do work to the ball)?

annAB · Accepted Answer

The tension of the string constitutes the centripetal force that allows the circular motion, and coincides at every instant with the radius of the circular motion, joining the ball with the trajectory centre.
Then at every instant of time the shift vector of the ball $vec{s}$ is perpendicular to the string tension $vec{T}$ (in a circular motion shift and radius are perpendicular to each other) and so the work $L= vec{T}cdot vec{s}=0$ at every time (scalar product of perpendicular vectors is null).

Why doesn't tension do any work in a vertical circular motion?

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