The Work Calculator finds mechanical work from force and displacement. When force and motion share a line, W = Fs. When force acts at an angle, W = Fs cosθ uses only the component parallel to the displacement. Results are in joules, with the substituted formula shown so a 50 N push through 3 m is visibly 150 J rather than a bare number.
Work is a transfer of energy. Positive work adds energy to the system along the chosen accounting; negative work removes it. That sign convention is what links this page to changes in kinetic energy through the work-energy theorem.
Calculate work from force and displacement
When a constant force acts in the same direction as the displacement, work is the product W = Fs. Force in newtons and displacement in metres give work in joules, because 1 J = 1 N·m. The Work Calculator multiplies the two magnitudes in the aligned-force mode and converts units on each field before multiplying.
| Symbol | Quantity | SI unit |
|---|---|---|
| W | work | joules (J) |
| F | force magnitude | newtons (N) |
| s | displacement magnitude | metres (m) |
Constant force is the assumption. A force that changes along the path needs an integral (or a graph of F against s, where work is the area under the curve). Introductory problems usually keep F steady so the product is enough.
Displacement is the straight-line change in position, not the path length when the path bends. Carrying a box around a room and ending where you started gives zero displacement and zero work by the weight if height never changed, even though arms feel tired for physiological reasons outside this definition.
Scientific notation in the 3.45e9 form is accepted. Large industrial forces and small laboratory displacements both fit without manual rescaling.
Calculate work when force acts at an angle
Forces often pull at an angle to the motion: a tow rope, a spreader handle, a suitcase strap. Only the component of force parallel to the displacement does work. With θ the angle between the force vector and the displacement vector, W = Fs cosθ. The Work Calculator's angled mode asks for F, s and θ and applies that factor.
| Case | θ | cosθ | Effect on work |
|---|---|---|---|
| Force along motion | 0° | 1 | Full Fs |
| Force at 60° | 60° | 0.5 | Half of Fs |
| Force perpendicular | 90° | 0 | Zero work |
| Force opposing motion | 180° | −1 | Negative work, magnitude Fs |
At θ = 60°, half the force contributes. At θ = 90°, none of it does. At angles above 90°, cosθ is negative and work is negative: the force component fights the displacement.
Resolve by components if that is clearer: F_parallel = F cosθ, then W = F_parallel × s. Same arithmetic. Report θ from the displacement direction consistently; measuring from the vertical by habit and calling it θ without adjusting is a common source of wrong signs in homework.
Apply the work-energy theorem
Net work on a particle equals the change in its kinetic energy: W_net = ΔKE = KE_f − KE_i. Speed up an object and net work is positive; slow it down and net work is negative.
The Work Calculator can frame a result as that change when you relate force to speed rather than stopping at Fs alone.
Net work means the sum of work from every force, or equivalently the work of the resultant force. Friction often contributes negative work while an applied push contributes positive work; the algebraic sum is what equals ΔKE.
Example link to kinetic energy: increasing a 2 kg mass from rest to 10 m/s requires ΔKE = 100 J − 0 = 100 J of net work. That matches the kinetic-energy fixture on the sibling page. If a constant net force does that work over 5 m, then F_net = W/s = 100/5 = 20 N.
Potential energy changes can be folded into the bookkeeping as work of gravity or as ΔPE on the other side of the ledger. Pick one convention and hold it. Mixing "work of gravity" and "ΔPE" in the same equation double-counts.
Calculate work for 50 N over 3 m
A constant 50 N force aligned with a 3 m displacement does 150 J of work. That pair is the worked fixture for this page, and the angled and opposing-force variants below show how the same magnitudes yield 75 J or −150 J when the cosine factor changes.
1. List what is known. F = 50 N, s = 3 m, force parallel to displacement (θ = 0°).
2. Write the formula. W = Fs (or W = Fs cos0° = Fs × 1)
3. Substitute. W = 50 × 3
4. Multiply. W = 150 J
If the same 50 N force acts at 60° to the displacement over the same 3 m:
W = 50 × 3 × cos60° = 150 × 0.5 = 75 J
If the force is 50 N opposite the motion (θ = 180°) over 3 m:
W = 50 × 3 × (−1) = −150 J
Negative work means 150 J was removed from the object's mechanical energy along that accounting. The calculator returns the signed value and labels the angle case so 150 J and −150 J are not confused.
Recognise when work is zero
Work is zero when the displacement is zero, or when the force is perpendicular to the displacement (cos90° = 0), or when the force itself is zero. The Work Calculator returns 0 in those cases rather than implying that "effort" occurred in the physics sense.
Pushing a wall as hard as you like does no mechanical work on the wall if the wall does not move. Muscle chemistry still expends metabolic energy; that is a different account. Introductory mechanics grades the mechanical definition.
Carrying a briefcase at constant height across a level floor, the upward force balances weight and is perpendicular to the horizontal displacement, so that supporting force does no work. A horizontal push that actually moves the case does work. Gravity does no work on the level path because the vertical displacement is zero.
Uniform circular motion at constant speed has centripetal force perpendicular to the instantaneous displacement along the path, so that centripetal force does no work and kinetic energy stays constant. Speed changes need a tangential force component.
Understand negative work
Negative work occurs when the force component opposes the displacement. Friction on a sliding block, drag on a vehicle and the force in a braking system typically do negative work: they remove kinetic energy. The Work Calculator keeps the sign so W = −40 J is readable as energy leaving the mechanical description of the object.
Braking a car: friction and brake forces act opposite the velocity. Displacement is forward, force is backward, θ = 180°, cosθ = −1, work is negative, and KE falls. The energy becomes heat in pads, discs and tyres.
Catching a ball: your hand applies a force opposite the ball's velocity while the ball displaces forward during the catch. Work by the hand on the ball is negative; the ball's KE drops to zero.
Gravity does negative work on an object thrown upward (force down, displacement up) and positive work on the way down. Over a closed vertical round trip with the same path up and down, the total work by gravity is zero, which matches returning to the same PE.
Distinguish work from power
Work is energy transferred; power is the rate of that transfer. Average power over an interval is P = W/t, and when force and velocity are aligned, P = Fv. The SI unit of power is the watt (1 W = 1 J/s).
The Work Calculator reports work; divide by time when a problem asks how fast that energy moved.
| Quantity | Symbol | SI unit | Meaning |
|---|---|---|---|
| Work | W | joule (J) | Energy transferred |
| Power | P | watt (W) | Joules per second |
Doing 150 J in 2 s is 75 W average power. Doing the same 150 J in 0.5 s is 300 W. Same work, different power. Engines and motors are rated in watts or horsepower because users care about delivery rate as much as total energy.
Kilowatt hours on an electricity bill are energy (work), not power. A kilowatt is power; multiplying by hours converts to energy. Keeping those labels straight avoids the common mix-up between a 2 kW heater and a 2 kWh energy use.
Frequently asked questions
What is the formula for work in physics?
For a constant force along the displacement, W = Fs. When the force is at an angle θ to the displacement, W = Fs cosθ. Force is in newtons, displacement in metres, work in joules. Variable forces need integration or the area under an F-s graph.
What is the work-energy theorem?
Net work on a particle equals its change in kinetic energy: W_net = ΔKE. Positive net work speeds the particle up; negative net work slows it down. This is the main bridge between the Work Calculator and the Kinetic Energy Calculator.
When is work zero?
Work is zero if displacement is zero, if force is zero, or if force is perpendicular to displacement. Pushing an immovable wall does no mechanical work on the wall. A centripetal force on uniform circular motion does no work.
Can work be negative?
Yes. When the force component opposes the displacement, cosθ is negative and W is negative. Friction and braking commonly do negative work, removing kinetic energy from the object.
How is work different from power?
Work is the energy transferred (joules). Power is how fast that energy is transferred (watts, or joules per second). The same work done in less time means greater power.
Does holding a heavy mass do work?
If the mass is stationary, displacement is zero and mechanical work on the mass is zero. Your muscles still consume metabolic energy to maintain tension. Physics courses mark the mechanical answer; biology classes discuss the metabolic cost separately.
What angle should be used in W = Fs cosθ?
Use the angle between the force vector and the displacement vector. If a problem gives the angle between a rope and the horizontal, and the displacement is horizontal, that angle is θ. If it gives the angle to the vertical, convert before substituting.
Is work a vector?
Work is a scalar. It can be positive or negative, but it has no direction of its own. Force and displacement are vectors; their dot product F·s produces the scalar work.
How does friction do work?
Kinetic friction acts opposite the sliding displacement, so its work is negative: W = −f_k s on a level path of length s. That negative work reduces mechanical energy. Static friction on a rolling object without slip can do zero work in idealized models because the contact point is instantaneously at rest.
Summary
The Work Calculator evaluates W = Fs for aligned force and displacement and W = Fs cosθ when force acts at an angle, returning joules with signed results when work is negative. The worked fixture 50 N through 3 m yields 150 J.
Net work equals the change in kinetic energy, zero work occurs for zero displacement or perpendicular force, and power measures the rate of doing work in watts rather than the work itself.