How to Calculate Work in Physics: Complete Guide
Master mechanical work W = Fd cos θ, the work–energy theorem, and power P = W/t. Worked examples cover lifts, ramps, and motors, with common mistakes and the Work and Power Calculator.
How to Calculate Work in Physics
In everyday speech, "work" means effort. In physics, work is a precise measure of energy transferred by a force acting through a displacement. Hold a heavy suitcase motionless and you may tire, but the physics work done on the suitcase is zero — no displacement, no work.
That definition links force, motion, and energy. Learn it once and you can analyse machines, efficiency, and power with the same language.
The core formula
When a constant force \vecF acts while an object undergoes displacement \vecd:
W = F d \cosθ
Where:
- W is work in joules (J); 1\,\mathrmJ = 1\,\mathrmN· m
- F is the magnitude of the force in newtons
- d is the magnitude of the displacement in metres
- θ is the angle between the force and the displacement vectors
In vector notation:
W = \vecF · \vecd
Only the component of force parallel to the displacement contributes. Perpendicular forces (ideal centripetal force on a circular path at constant speed) do no work.
Positive, negative, and zero work
Gravity does positive work when something falls and negative work when something is lifted at constant speed by you (you do positive work; gravity's work is negative).
Work–energy theorem
The net work on a particle equals its change in kinetic energy:
W_\mathrmnet = Δ KE = (1)/(2)mv_f^2 - (1)/(2)mv_i^2
This is often the fastest path to a speed or a stopping distance once you know the forces.
Power
Power is the rate of doing work:
P = (W)/(t)
For a constant force parallel to constant velocity \vecv:
P = F v
Unit: watt (W), where 1\,\mathrmW = 1\,\mathrmJ/s. Do not confuse the symbol W for work with W for watts — context distinguishes them; many texts use P for power and spell out joules.
Average horsepower in US customary marketing is often compared via 1\,\mathrmhp ≈ 746\,\mathrmW.
Worked example: pushing a crate
Problem. You push a crate 5.0 m across a floor with a 60 N force at 25° above the horizontal. How much work do you do?
W = (60)(5.0)\cos 25° = 300 × 0.9063 = 272 \text J
The vertical component does not contribute to work along the horizontal displacement (it may reduce normal force and friction, which is a separate force's work).
Worked example: lifting at constant speed
Problem. Lift a 15 kg box vertically 1.2 m at constant speed. How much work do you do? Take g = 9.81\,\mathrmm/s^2.
At constant speed, your upward force equals weight: F = mg.
W = mgh = 15 × 9.81 × 1.2 = 177 \text J
That work becomes gravitational potential energy mgh. Net work is zero (your work +mgh, gravity -mgh), so Δ KE = 0, as expected.
Worked example: kinetic friction stopping a sled
Problem. A 20 kg sled sliding at 4.0 m/s on level snow is stopped by kinetic friction \mu_k = 0.10. Find the stopping distance. (g = 9.81)
Friction force: f_k = \mu_k mg = 0.10 × 20 × 9.81 = 19.62\,\mathrmN, opposite displacement, so W_\mathrmfric = -f_k d.
Work–energy:
-f_k d = 0 - (1)/(2)mv_i^2
d = (mv_i^2)/(2 f_k) = (20 × 16)/(2 × 19.62) = (320)/(39.24) = 8.15 \text m
Worked example: power of a motor
Problem. A hoist lifts a 500 kg load at a steady 0.40 m/s. What useful power does it deliver?
F = mg = 500 × 9.81 = 4905\,\mathrmN
P = Fv = 4905 × 0.40 = 1962\,\mathrmW ≈ 1.96\,\mathrmkW
If the motor is 80% efficient, electrical input power is about 1962 / 0.80 = 2450\,\mathrmW.
Variable force and area under the curve
When force varies with position, work is the integral:
W = ∫_x_1^x_2 F_x\,dx
Graphically, that is the area under F_x versus x. For a spring from x = 0 to x = A:
W = ∫_0^A kx\,dx = (1)/(2)kA^2
which matches the stored elastic energy.
Work by gravity and path independence
Gravitational work between two heights depends only on Δ h:
W_g = -mgΔ h
(with Δ h = h_f - h_i). Path shape does not matter for gravity or ideal springs — they are conservative. Friction is not: sliding around a long path dissipates more energy.
Machines, mechanical advantage, and efficiency
Simple machines trade force and distance. Ideal work in equals work out:
F_\mathrmin d_\mathrmin = F_\mathrmout d_\mathrmout
Real efficiency:
\eta = \fracW_\mathrmoutW_\mathrmin < 1
A ramp lets you use smaller F_\mathrmin over larger d_\mathrmin to gain the same mgh.
Applications
Vehicle braking. Brake friction does negative work equal to the loss of kinetic energy (plus some heat in tires and air).
Human metabolism. Physics work against gravity on a hike is mgΔ h; metabolic energy cost is larger because muscles are not 100% efficient.
Industrial motors. Spec sheets list rated power; torque and speed set how that power is delivered (P = \tau\omega).
Crash analysis. Work by crumple zones equals the kinetic energy that must be removed — longer crush distance lowers average force.
Common mistakes
Equating effort with work. Static holds: d = 0, so W = 0 on the object.
Dropping \cosθ. Angled forces need the parallel component.
Using weight in the wrong place. W_\mathrmlift = mgh for vertical lifts at constant speed; do not multiply by an extra g.
Confusing work W with power in watts. Power needs a time (or speed).
Ignoring the sign. Negative work reduces mechanical energy; omit the sign and stopping-distance problems fail.
Adding path length for gravity work. Only vertical change matters for W_g.
Frequently asked questions
Can work be done with no net force? Individual forces can do work even when net force is zero. Carry a box at constant velocity: you do positive work against gravity and friction may do negative work; net work can be zero while your muscles still expend chemical energy.
Is energy the same as work? Work is one way to transfer energy. Heat transfer is another. Energy is a property of a system; work is a process quantity.
Why is the unit of work the same as torque (N·m)? Same dimensions; different meaning. Work uses force along displacement; torque uses force with lever arm. Writing joules for energy and N·m for torque avoids confusion.
How do I handle multiple forces? Compute work for each force and add (scalar sum), or find the net force's work — both equal Δ KE when all forces on the particle are included.
What is "useful work"? The portion converted to the desired form (lift height, output shaft energy). The rest is usually heat. Efficiency is useful output over total input.
Does centripetal force do work? For uniform circular motion, the centripetal force is perpendicular to velocity every instant, so its work is zero and speed stays constant.
Summary
Mechanical work by a constant force is:
W = F d \cosθ
Net work changes kinetic energy; gravity and springs also store potential energy. Power is work per time, P = W/t or P = Fv when force and velocity align. Keep the angle, the sign, and the distinction between work and power clear.
Calculate work and power with our Work and Power Calculator.