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I know that work done by a constant force is $W=\vec{F}\cdot \vec{x}$, and that work done by a variable force is $\int \vec{F} d\vec{x}$. However, how do you calculate work done by a varying angle?

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  • $\begingroup$ Are you sure you don't mean torque? Torques & angles are analogous to forces and distances, in that order. $\endgroup$
    – TimWescott
    Oct 10, 2019 at 0:21

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Indeed, it is the same formula.

$$W=\int_C \vec{F}\cdot d\vec{x} $$

Check that this is the same as

$$\int_C |\vec{F}|\cdot \cos(\alpha)\cdot dx $$

This integral is a path integral, that means that you're summing all the contributions along all the tiny portions of the path of your trajectory.

Both $F$ or the angle can vary along the path. Anythin inside the integral can.

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There is a difference between $\int Fdx$ and $\int \vec{F}.d\vec{x}$.Basically a difference of dot product which is capable of calculating work by a variable angle.

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