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The work done by a force \( \mathbf{F} \) along a path \( C \) is

  • A.\( \displaystyle\int_{C} |\mathbf{F}|\,ds \)
  • B.\( \displaystyle\int_{C} \mathbf{F}\,ds \)
  • C.\( \displaystyle\int_{C} \mathbf{F} \cdot d\mathbf{r} \)
  • D.\( \displaystyle\int_{C} \mathbf{F} \times d\mathbf{r} \)

Answer: C

Only the component of the force along the motion does work, so each element contributes \( \mathbf{F} \cdot d\mathbf{r} \) and

\[ W \;=\; \int_{C} \mathbf{F} \cdot d\mathbf{r}. \]

Integrating the magnitude alone would count a sideways force as though it were pushing along the path, and a cross product would give a vector rather than an amount of work

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