Work Done by a Force

• The physics definition of work and why it differs from everyday meaning
• How angle between force and displacement determines work done
• When work is positive, negative, or zero
• How to calculate work at an angle using W = F·d·cosθ

The Physics Definition of Work In everyday language, "work" means any physical or mental effort. In physics, work has a precise, narrow meaning:

W = F × d × cos θ

where:

• W = work done (Joules, J)
• F = magnitude of the applied force (Newtons, N)
• d = magnitude of displacement (metres, m)
• θ = angle between the force vector and the displacement vector

Work is done on an object only when:

1. A force acts on it, AND
2. The object moves (displacement is non-zero), AND
3. The force has a component along the direction of displacement (θ ≠ 90°)

If any one of these conditions is missing, no work is done in the physics sense.

The angle θ and its effect

θ = 0° (force parallel to displacement): W = Fd·cos0° = Fd·1 = Fd → maximum positive work Example: pushing a cart horizontally with a horizontal force.

θ = 90° (force perpendicular to displacement): W = Fd·cos90° = Fd·0 = 0 → zero work Example: carrying a heavy bag horizontally — your arms exert an upward force (to support the bag), but the displacement is horizontal. The force is vertical, motion is horizontal, θ = 90°, so W = 0. Your muscles still fatigue because they do internal biological work, but mechanically zero work is done on the bag in the horizontal direction.

θ = 180° (force opposite to displacement): W = Fd·cos180° = Fd·(−1) = −Fd → negative work Example: friction opposes motion. When a box slides forward, friction acts backward (θ = 180°). The work done by friction is negative — it removes energy from the system.

Positive, negative, and zero work

• Positive work: the force aids the motion, adding energy to the object (e.g., engine force on a car).
• Negative work: the force opposes the motion, removing energy (e.g., friction, air resistance, brakes).
• Zero work: force is perpendicular to displacement (e.g., normal force on a moving object, gravity on horizontal motion).

Units of work Work is measured in Joules (J). 1 J = 1 N·m = the work done by a 1 Newton force moving an object 1 metre in the direction of the force.

Work-Energy Theorem The net work done on an object equals the change in its kinetic energy:

W_net = ΔKE = (1/2)mv² − (1/2)mu²

This connects the concept of work to motion and is a powerful tool in problem-solving.

Problem: A person pushes a 10 kg box across a floor a distance of 5 m. The push force is 20 N directed at 30° below the horizontal. Find the work done by the person.

Solution:

The force is 20 N at 30° to the horizontal (displacement direction).

W = F × d × cos θ W = 20 × 5 × cos 30° W = 100 × (√3/2) W = 100 × 0.866 W ≈ 86.6 J

Note: The angle between the applied force and the displacement is 30° (not 90° — the person is pushing forward-and-downward, but the box moves forward). The work done is approximately 86.6 J.

If the person had pushed horizontally (θ = 0°): W = 20 × 5 × cos 0° = 20 × 5 × 1 = 100 J

The angled push does less useful work on the horizontal displacement than a purely horizontal push would.

• Thinking physical effort equals physics work. Holding a 20 kg weight stationary overhead is exhausting, but because there is no displacement, W = 0. Physics only counts work when displacement occurs.
• Using the full force magnitude when force is at an angle. Always use F cosθ (the component of force along displacement), not F alone.
• Confusing positive and negative work. Work done by friction is always negative (it opposes motion). Total work can be negative if the opposing forces dominate.
• Ignoring the direction of displacement for θ. θ is measured between the force vector and the displacement vector, not between the force and the horizontal or any other reference.

1. A force of 50 N acts horizontally on a box, moving it 4 m horizontally. Find the work done.
2. A 500 g book is lifted vertically 1.2 m with a steady upward force equal to its weight (g = 10 m/s²). What is the work done by the lifting force? What is the work done by gravity?
3. A refrigerator is pushed across a floor for 3 m with a 200 N force at 45° to the horizontal. How much work is done?