What is the work done by a constant force F on a displacement d when the angle between force and displacement is theta?

What is the work done by a constant force F on a displacement d when the angle between force and displacement is theta?

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

Answer: (C)

Work is the energy transferred by a force as it moves an object along a displacement. For a constant force, the work equals the component of the force in the direction of the displacement times the distance moved. That component is F cos theta, so the work is W = (F cos theta) d = F d cos theta. This comes from the dot product F · d, which automatically picks out how much of the force actually acts along the path. If the force points in the same direction as the displacement (theta = 0), cos theta = 1 and W = F d. If the force is perpendicular (theta = 90°), cos theta = 0 and no work is done. If it points opposite the motion (theta = 180°), cos theta = -1 and the work is negative. Thus the expression with cos theta is the correct one.

Work is the energy transferred by a force as it moves an object along a displacement. For a constant force, the work equals the component of the force in the direction of the displacement times the distance moved. That component is F cos theta, so the work is W = (F cos theta) d = F d cos theta. This comes from the dot product F · d, which automatically picks out how much of the force actually acts along the path.

If the force points in the same direction as the displacement (theta = 0), cos theta = 1 and W = F d. If the force is perpendicular (theta = 90°), cos theta = 0 and no work is done. If it points opposite the motion (theta = 180°), cos theta = -1 and the work is negative.

Thus the expression with cos theta is the correct one.