Work and conservation of energy: (b) W = Fx cos i for work done by a force

The idea

Work done by a force is the energy transferred when the force moves its point of application. The equation W = Fx cos θ gives the work done when a constant force F acts at an angle θ to the displacement x. The cos θ factor selects the component of force along the displacement. If θ = 0°, cos 0° = 1, so W = Fx. If θ = 90°, cos 90° = 0, so no work is done by that force. If θ = 180°, cos 180° = −1, so W = −Fx, meaning energy is transferred away from the object. Work is a scalar quantity measured in joules (J); 1 J = 1 N m.

For example: Pulling a box with a 50 N force at 30° to a 4.0 m horizontal displacement gives W = 50 × 4.0 × cos 30° ≈ 173 J. Always resolve the force along the displacement before multiplying.

What earns the marks

• State that work done equals the product of force, displacement and the cosine of the angle between them: W = Fx cos θ.
• Identify θ as the angle between the force vector and the displacement vector, not the angle to the horizontal unless that is the same direction.
• Use the component of force along the displacement: F cos θ, then multiply by displacement x.
• Recognise that when θ = 90°, cos 90° = 0, so the force does no work on the object.
• Recognise that when θ = 180°, cos 180° = −1, so the work done is negative and energy is transferred away.
• Calculate work in joules using consistent SI units: force in newtons, displacement in metres.
• Substitute values correctly, including evaluating cos θ in degrees or radians as appropriate.

Common mistakes and how to put them right

• Using the full force F instead of the component F cos θ when the force is at an angle to the displacement.

  Put it right by resolving along the displacement first.

• Taking θ as the angle between the force and the horizontal rather than the angle between the force and the displacement.

  Put it right by identifying the displacement direction and measuring the angle from it.

• Forgetting that cos 90° = 0, so a force perpendicular to motion does no work.

  Put it right by checking the angle before calculating.

• Treating negative work as impossible or as a mistake.

  Put it right by interpreting W = −Fx as energy transferred away from the object, for example by friction.

• Mixing units, such as using force in N and displacement in cm without converting.

  Put it right by converting all lengths to metres before multiplying.

Exam tips

• Sketch the force and displacement vectors and mark the angle θ between them before substituting into W = Fx cos θ.
• Check whether the force is constant and the displacement is along a straight line; the equation applies to a constant force over a straight-line displacement.
• If the force and displacement are in the same direction, use W = Fx; if perpendicular, state that no work is done by that force.
• Keep the sign of cos θ: a negative result means energy is transferred away from the object, not that the calculation is wrong.