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Why do radial forces do no work?

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Recall the definition of work done, $W$ by a force, $\textbf{F}$, over a change in displacement $\textbf{x}$

\begin{equation} W = \textbf{F}\cdot\textbf{x}. \end{equation}

In a circular orbit, the force is always perpendicular to the change in displacement (i.e. radial lines are always perpendicular to tangents) and the scalar product therefore ensures that the gravitational force does no work on a body in a circular orbit.

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Radial forces do work. For example gravitation.

Radial forces where $\vec{r} \cdot \hat v = 0$ do no work.

$W = \int \vec{F} \cdot \vec{dr}$

$\int f(x,y,z) \hat r \cdot \vec{dr}$

$\int f(x,y,z) \hat r \cdot \vec{v}dt$

$\int [\hat r \cdot \vec{v}] f(x,y,z)dt$

Hence if the motion of an object is always perpendicular to the force, ie $[\hat r \cdot \vec{v}] = 0$, then the work done is zero.

An example of this is circular motion

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