Work Calculator
Calculate the work done by a force over a distance using the formula W = F×d×cos(θ). Enter the force magnitude, displacement, and angle between force and motion to find the work in joules. See also our Force Calculator and Kinetic Energy Calculator.
How to Calculate Work
In physics, work is defined as the transfer of energy that occurs when a force causes an object to move through a displacement. Unlike the everyday meaning of "work," the physics definition is precise: work is done only when a force component acts along the direction of motion. Holding a heavy box stationary does no work in the physics sense (no displacement), even though it feels tiring. The concept of work was formalized in the 19th century and is fundamental to thermodynamics and mechanics.
To calculate work, use the formula W = F×d×cos(θ), where F is the magnitude of the force, d is the displacement (distance moved), and θ is the angle between the force vector and the displacement vector. When the force is in the same direction as motion (θ = 0°), cos(0°) = 1 and W = F×d. When the force is perpendicular to motion (θ = 90°), cos(90°) = 0 and no work is done. When the force opposes motion (θ = 180°), work is negative (energy is removed from the object).
The SI unit of work is the joule (J), equal to one newton-meter (N⋅m). One joule is the work done when a force of 1 newton moves an object 1 meter in the direction of the force. Other common units include the kilowatt-hour (1 kWh = 3,600,000 J), the calorie (1 cal = 4.184 J), the foot-pound (1 ft⋅lbf = 1.356 J), and the electron-volt (1 eV = 1.602×10⁻¹⁹ J).
The work-energy theorem connects work to kinetic energy: the net work done on an object equals its change in kinetic energy (W_net = ΔKE). This powerful principle allows solving many mechanics problems without tracking forces at every instant. Work is also related to power: power is the rate of doing work (P = W/t = F×v). Understanding work is essential for analyzing machines, engines, and energy systems.
Work Formula
Work (constant force, straight line):
W = F × d × cos(θ)
Work (force parallel to motion):
W = F × d (when θ = 0°)
Work (variable force):
W = ∫F⋅dx (integral of force over displacement)
Work-Energy Theorem:
W_net = ΔKE = ½mv₂² - ½mv₁²
Work against gravity:
W = mgh (lifting height h)
Work by spring:
W = ½kx² (compressing/stretching by x)
Power (rate of work):
P = W/t = F×v×cos(θ)
Example Calculation
A person pushes a box with 50 N of force over 10 meters at an angle of 30° below horizontal:
Given: F = 50 N, d = 10 m, θ = 30°
W = F×d×cos(θ) = 50 × 10 × cos(30°)
W = 50 × 10 × 0.8660 = 433.01 J
Horizontal force component: 50 × cos(30°) = 43.30 N
Vertical force component: 50 × sin(30°) = 25.00 N
If done in 5 seconds:
Power = W/t = 433.01/5 = 86.60 W
Compare: at θ = 0°, W = 50 × 10 × 1 = 500 J
At θ = 60°, W = 50 × 10 × 0.5 = 250 J
At θ = 90°, W = 50 × 10 × 0 = 0 J (no work!)
Work Reference Table
| Force (N) | Distance (m) | Angle (°) | Work (J) |
|---|---|---|---|
| 1 N | 1 m | 0° | 1 J |
| 10 N | 5 m | 0° | 50 J |
| 50 N | 10 m | 0° | 500 J |
| 100 N | 10 m | 0° | 1000 J |
| 50 N | 10 m | 30° | 433 J |
| 50 N | 10 m | 45° | 354 J |
| 50 N | 10 m | 60° | 250 J |
| 50 N | 10 m | 90° | 0 J |
| 200 N | 50 m | 0° | 10000 J |
| 500 N | 100 m | 0° | 50000 J |
Frequently Asked Questions
What is work in physics?
In physics, work is the energy transferred to or from an object by a force acting through a displacement. It is calculated as W = F×d×cos(θ), where θ is the angle between force and displacement. Work is positive when force has a component in the direction of motion (energy added), negative when opposing motion (energy removed), and zero when force is perpendicular to motion. The SI unit is the joule (J). Work is a scalar quantity — it has magnitude but no direction.
Can work be negative?
Yes. Work is negative when the force has a component opposing the direction of motion (angle between 90° and 180°). Friction always does negative work on a sliding object because it opposes motion. When you lower a box slowly, gravity does positive work (force and motion both downward) while your arms do negative work (your force is upward, motion is downward). Negative work means energy is being removed from the object — its kinetic energy decreases.
Why does carrying a box horizontally do no work?
When you carry a box horizontally at constant height, the force you exert is vertical (upward, supporting the weight) while the displacement is horizontal. Since the angle between force and displacement is 90°, and cos(90°) = 0, the work done on the box is zero. Your muscles do metabolic work (burning calories) to maintain the force, but no mechanical work is transferred to the box. The box's kinetic energy doesn't change, confirming zero net work.
What is the work-energy theorem?
The work-energy theorem states that the net work done on an object equals its change in kinetic energy: W_net = ΔKE = ½mv₂² - ½mv₁². This is one of the most powerful principles in mechanics. It means that if you know the net work done on an object, you can find its speed change without knowing the details of the forces at every instant. It applies to any situation — constant or variable forces, straight or curved paths — making it extremely versatile for problem-solving.
What is the difference between work and power?
Work measures the total energy transferred (in joules), while power measures the rate of energy transfer (in watts = joules/second). Two people can do the same work lifting identical boxes to the same height, but the one who does it faster exerts more power. P = W/t, or equivalently P = F×v (force times velocity). A 100 W light bulb converts 100 joules of electrical energy to light and heat every second. A car engine producing 150 kW can do 150,000 joules of work per second.
How is work related to potential energy?
Work done against a conservative force (like gravity or a spring) is stored as potential energy. Lifting an object height h against gravity requires work W = mgh, which is stored as gravitational PE. Compressing a spring by distance x requires work W = ½kx², stored as elastic PE. This stored energy can later be converted back to kinetic energy. The relationship W = -ΔPE connects work by conservative forces to potential energy changes. Non-conservative forces (like friction) convert mechanical energy to thermal energy irreversibly.
Work in Real-World Applications
The concept of work is central to understanding machines and energy systems. A simple machine (lever, pulley, inclined plane) doesn't reduce the work needed — it reduces the force required by increasing the distance. A ramp lets you lift a heavy object with less force over a longer distance, but the total work (force × distance) remains the same (ignoring friction). This is the principle of conservation of energy applied to machines.
In thermodynamics, work appears as pressure-volume work (W = P×ΔV) when gases expand or compress. Internal combustion engines convert chemical energy to mechanical work through gas expansion. Refrigerators and heat pumps require work input to move heat from cold to hot reservoirs. The efficiency of any engine is defined as the ratio of useful work output to total energy input, and the second law of thermodynamics sets fundamental limits on this efficiency.