Which equation expresses work done by a constant force when displacement and force form an angle theta?

Which equation expresses work done by a constant force when displacement and force form an angle theta?

• A. W = F d cos theta
• B. W = F d sin theta
• C. W = F d^2 cos theta
• D. W = F / d cos theta

Answer: (A)

Work is the transfer of energy due to motion under a force, and with a constant force only the part of that force that points along the displacement does work. The amount of work is the component of the force in the direction of motion multiplied by the distance moved, which is F cos theta times d. In other words, W = F d cos theta, the dot product of the force and displacement vectors. If the force points the same way as the motion (theta = 0), you get W = Fd. If the force is perpendicular to the motion (theta = 90°), no work is done (W = 0). If it points opposite to motion (theta = 180°), the work is negative (W = -Fd). The other forms don’t fit: sin theta would involve the perpendicular component that doesn’t contribute to work; d^2 would have the wrong units and misrepresents the relationship; dividing by d also breaks the dimensional consistency.

Work is the transfer of energy due to motion under a force, and with a constant force only the part of that force that points along the displacement does work. The amount of work is the component of the force in the direction of motion multiplied by the distance moved, which is F cos theta times d. In other words, W = F d cos theta, the dot product of the force and displacement vectors.

If the force points the same way as the motion (theta = 0), you get W = Fd. If the force is perpendicular to the motion (theta = 90°), no work is done (W = 0). If it points opposite to motion (theta = 180°), the work is negative (W = -Fd).

The other forms don’t fit: sin theta would involve the perpendicular component that doesn’t contribute to work; d^2 would have the wrong units and misrepresents the relationship; dividing by d also breaks the dimensional consistency.