Work Done in Physics: Formula & Worked Examples

You can hold a heavy box for an hour and get exhausted — but in the physics sense, you've done zero work on the box. No displacement, no work. That's not a trick definition; it's a precise one that makes the maths consistent. Work in physics is force multiplied by displacement in the direction of that force, and understanding what that means physically is more important than memorising the formula.

Work is the link between force and energy. The work-energy theorem — that the net work done on an object equals its change in kinetic energy — is one of the most useful shortcuts in mechanics. It lets you bypass force analysis entirely in many problems and jump straight from initial to final state.

What Is Work Done in Physics?

Work done is the energy transferred when a force causes an object to move through a displacement. The key requirement: the object must actually move, and the force (or a component of it) must be in the direction of movement.

Formally: W = F · d = Fd cos θ, where θ is the angle between the force vector and the displacement vector. This is the scalar (dot) product of force and displacement.

The unit is the joule (J). 1 J = 1 N·m = 1 kg·m²·s⁻². One joule is the work done when a force of 1 newton moves an object 1 metre in the direction of the force.

The W = Fd cos θ Formula — Three Cases

Case 1: θ = 0° (force parallel to displacement)
cos 0° = 1, so W = Fd. This is the simplest case — pushing a box horizontally, a car engine propelling a car forward. All the force contributes to work.

Case 2: 0° < θ < 90° (force at an angle to displacement)
Only the component of force along the displacement (F cos θ) does work. The component perpendicular to displacement (F sin θ) does no work. Example: dragging a suitcase with a handle at 30° to the floor.

Case 3: θ = 90° (force perpendicular to displacement)
cos 90° = 0, so W = 0. A centripetal force, the normal force on a flat surface, the magnetic force on a moving charge — all do zero work because they're perpendicular to motion. This is why magnetic forces never change a particle's kinetic energy.

Case 4: 90° < θ ≤ 180° (force opposing displacement)
cos θ is negative, so W is negative. The force removes energy from the object. Friction on a sliding box does negative work — it takes kinetic energy and converts it to heat.

Worked Example 1: Simple Horizontal Push

A 15 N force pushes a crate 6.0 m along a flat floor. Force and displacement are parallel (θ = 0°). Find the work done.

Worked Example 2: Force at an Angle

A child pulls a toy wagon 5.0 m with a rope tension of 20 N. The rope makes a 25° angle with the horizontal. Find the work done by the tension.

If the child held the rope horizontally (θ = 0°), work would be 100 J. The 25° angle reduces effective forward force to 20 cos 25° = 18.1 N — 9.4% less useful, the rest going into the ground as a downward component.

Worked Example 3: Lifting Against Gravity

A 5.0 kg box is lifted 1.2 m vertically at constant velocity. Find the work done by the lifting force.

At constant velocity, lifting force = weight = mg = 5.0 × 9.8 = 49 N. Force and displacement are both upward (θ = 0°):

This work is stored as gravitational potential energy: ΔPE = mgh = 5.0 × 9.8 × 1.2 = 58.8 J. ✓ Work done lifting = gravitational PE gained — they must be equal since KE doesn't change at constant velocity.

Worked Example 4: Work Done Against Friction

A 10 kg box slides 3.0 m across a floor. Coefficient of kinetic friction μ_k = 0.3. Find: (a) friction force, (b) work done by friction, (c) work done by the normal force.

(a) N = mg = 10 × 9.8 = 98 N; friction f = μ_k N = 0.3 × 98 = 29.4 N

(b) Friction opposes motion (θ = 180°): W_friction = fd cos 180° = 29.4 × 3.0 × (−1) = −88.2 J

(c) Normal force is perpendicular to motion (θ = 90°): W_normal = Nd cos 90° = 0 J

The −88.2 J means friction removed 88.2 J of kinetic energy from the box — converted to internal energy (heat) in the surfaces.

The Work-Energy Theorem

The work-energy theorem states that the net work done on an object equals the change in its kinetic energy:

This follows from Newton's second law: F = ma, W = Fd = mad, and using v² = u² + 2ad (SUVAT) gives W = ½mv² − ½mu².

It means: if you know all the forces acting and their displacements, you can find the final speed without needing to solve equations of motion at every instant. It's often the fastest method for problems involving forces and motion.

Work Done by Multiple Forces

When multiple forces act on an object, calculate the work done by each force separately, then sum them:

Alternatively, find the net force first, then calculate W = F_net × d (if all forces are constant). The net work then equals ΔKE by the work-energy theorem.

Work Done in a Variable Force

When force varies with position (like a spring: F = kx), W = Fd doesn't apply directly. Work is the area under the force-displacement graph:

For a spring compressed by x₀: W = ∫₀^x₀ kx dx = ½kx₀². This stored energy is elastic potential energy — the energy that fires an arrow when you release a bowstring.

Real-World Applications

Elevators: an elevator motor does work W = mgh to raise passengers. For a 600 kg elevator rising 20 m: W = 600 × 9.8 × 20 = 117,600 J = 117.6 kJ. A modern elevator motor with regenerative braking converts most of the descending work back to electricity.

Braking distance: when a car brakes, friction does negative work, removing kinetic energy. KE = ½mv². Using W_friction = −fd: d = mv²/(2f) = mv²/(2μmg) = v²/(2μg). This shows braking distance is proportional to v² — doubling speed quadruples stopping distance. At 30 mph (13.4 m/s) with μ = 0.7: d = 13.4²/(2 × 0.7 × 9.8) = 13.1 m. At 60 mph: 52.4 m — four times as far.

Mechanical advantage: simple machines (levers, pulleys, ramps) redistribute work. A ramp of length L and height h lets you apply force F = mgh/L over distance L instead of mg over distance h. Work input = work output (ignoring friction): FL = mgh. You trade force for distance — the machine does the same work, you just apply less force over more distance.

For related calculations, use the work-power calculator. For how work connects to kinetic energy, see the kinetic energy article, and for work done against gravity, see gravitational force.

Work Done and Energy Conservation

Work done is the mechanism by which energy is transferred between objects. The principle of conservation of energy says total energy is conserved — it can only change form, not appear or disappear. Work done by external forces converts energy between types:

• Work done lifting = gravitational PE gained: W = mgh
• Work done accelerating = kinetic energy gained: W = ½mv² − ½mu²
• Work done against friction = thermal energy (heat) generated: W_friction = fd (dissipated as internal energy)
• Work done compressing a spring = elastic PE stored: W = ½kx²

If all forces are conservative (gravity, springs — no friction), total mechanical energy (KE + PE) is conserved. Friction introduces non-conservative (dissipative) forces that convert mechanical energy to heat, meaning mechanical energy is not conserved — but total energy including heat still is.

Common Mistakes with Work Done

Multiplying the full force by distance when force is at an angle. If you pull a box along the floor with a rope at 30° and use W = Fd without the cos 30° factor, you overestimate work by 15%. Always use W = Fd cos θ.

Assuming stationary objects have no forces doing work. A wall pushes on a stationary box with a normal force, but since displacement = 0, work done = 0. Work requires both force and displacement. Holding a heavy weight motionless above your head requires significant muscular effort but does zero mechanical work (though your muscles do biochemical work internally).

Adding work done by all forces when the question asks for net work. Net work = work done by net force = ΔKE. If you push a box across the floor at constant velocity, your force does positive work and friction does equal negative work — net work = 0, consistent with ΔKE = 0 (no change in speed).

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