Which statement best describes the sign of work when the force is at an angle θ to the displacement?

Which statement best describes the sign of work when the force is at an angle θ to the displacement?

• A. The work is always positive for any angle θ.
• B. The work is always negative for any angle θ.
• C. The work is zero for any angle θ that is not 0.
• D. The work is W = Fd cos θ, and its sign depends on θ.

Answer: (D)

The key idea is that work is the component of the force in the direction of the displacement, times how far the object moves. This is written as W = F · d = F d cos θ, where θ is the angle between the force and the displacement. Because cos θ can be positive, zero, or negative (for angles between 0 and 180 degrees), the sign of the work changes accordingly. If the force points partly in the same direction as the motion (acute angle), cos θ is positive and the work is positive. If the force is perpendicular to the motion (θ = 90°), cos θ = 0 and no work is done by that force. If the force points opposite to the motion (obtuse angle), cos θ is negative and the work is negative. So the formula W = F d cos θ not only gives the magnitude of the work but also its sign based on θ. The statements claiming work is always positive or always negative are incorrect, and zero work occurs specifically when θ = 90°.

The key idea is that work is the component of the force in the direction of the displacement, times how far the object moves. This is written as W = F · d = F d cos θ, where θ is the angle between the force and the displacement.

Because cos θ can be positive, zero, or negative (for angles between 0 and 180 degrees), the sign of the work changes accordingly. If the force points partly in the same direction as the motion (acute angle), cos θ is positive and the work is positive. If the force is perpendicular to the motion (θ = 90°), cos θ = 0 and no work is done by that force. If the force points opposite to the motion (obtuse angle), cos θ is negative and the work is negative.

So the formula W = F d cos θ not only gives the magnitude of the work but also its sign based on θ. The statements claiming work is always positive or always negative are incorrect, and zero work occurs specifically when θ = 90°.