Work Calculator
Calculate work from force, displacement, and angle, or from change in kinetic energy using mass and speeds. Choose what you know and solve.
Work Calculator
Result will appear here...
What the work calculator does
In physics, work is energy transferred by a force, and there are two ways to measure it. One is from the force and the distance it acts over. The other is from the change in an object's motion, the energy its speed gained or lost. This calculator does both: it finds work from a force, a displacement, and the angle between them, or from a mass and its change in speed, and it can solve for any of the pieces.
Below is what work really means, the equations behind each method, when a force does no work at all, and a worked example.
How to use it
- Choose how to calculate work: from force and displacement, or from a velocity change.
- Choose what to find within that method, then enter the known values. The force method uses force, displacement, and the angle between them; the velocity method uses mass with the initial and final speeds.
- Press Calculate for the answer, or Reset to clear it.
What work means in physics
Work has a precise meaning here, narrower than the everyday word. It is done when a force moves something through a distance, and it measures the energy that force transfers. Push a box across the floor and you do work on it; the energy you spend goes into the box and whatever resists it. No movement means no work, however hard you strain: holding a heavy weight still, your muscles tire, but in the physics sense you do no work on the weight, because it does not move.
Work is the bridge between force and energy. A force on its own does not change an object's energy, but a force acting over a distance does, and that transferred energy is the work. It is measured in joules, the same unit as every other kind of energy.
The equations it uses
The force method uses the force F, the displacement d, and the angle θ between the direction of the force and the direction of motion:
W = F d cosθ
The cosine picks out the part of the force that lies along the motion, since only that part does work. The velocity method uses the work-energy theorem, where the work done equals the change in kinetic energy, with m for mass, u for the initial speed, and v for the final speed:
W = ½ m ( v² − u² )
Each can be rearranged to solve for any of its quantities, so you can find the force or displacement that a known work implies, or the speed an object reaches after a given amount of work.
When a force does no work
The angle in the force equation carries a surprising lesson. When the force points along the motion, the angle is zero and its cosine is one, so all of the force does work. When the force is at an angle, only part of it counts. And when the force is exactly sideways to the motion, at 90 degrees, the cosine is zero, and the force does no work at all, however strong it is.
This is why the tension in a string does no work on a ball swung in a circle: the pull is always sideways to the motion, so the ball's speed never changes from it. It is also why carrying a heavy bag across level ground does no work against gravity, since gravity pulls straight down while you move horizontally. A force only does work to the extent it has a share along the direction of travel.
Work and the work-energy theorem
The two methods are not separate facts, they are two sides of one idea. The work-energy theorem says that the total work done on an object equals the change in its kinetic energy. Do positive work on something and you speed it up; do negative work, like friction or braking, and you slow it down. The energy you put in with a force over a distance shows up exactly as a change in the energy of motion.
That is why this calculator can find work either way and get the same answer. Push a box with a known force over a known distance, and the work you did equals the kinetic energy the box gained, provided nothing else drained it away. The force-and-distance view and the change-in-speed view are simply two ways of measuring the same transferred energy.
Units and precision
The calculator works in SI units: force in newtons, displacement in metres, mass in kilograms, speeds in metres per second, and the angle in degrees. Work comes out in joules. Results are rounded to a couple of decimal places for display, while the calculation itself runs at full precision.
A worked example
Suppose you push a box with a force of 50 N over a distance of 3 metres, pushing straight along the direction it moves, so the angle is 0.
The work done is W = 50 × 3 × cos0 = 50 × 3 × 1 = 150 J. Push instead at 60 degrees to the motion and only half the force counts, giving 50 × 3 × 0.5 = 75 J. And by the work-energy theorem, that 150 J of work, with nothing resisting, becomes 150 J of kinetic energy in the box.
Questions people ask
What is the formula for work?
Work equals force times displacement times the cosine of the angle between them, W = Fd cosθ. It can also be found as the change in kinetic energy, W = ½m(v² − u²).
Is work done if nothing moves?
No. Work requires movement through a distance. Holding a weight still takes effort but does no work on the weight in the physics sense, because it does not move.
Can a force do no work even while acting?
Yes. A force at right angles to the motion does no work, since the cosine of 90 degrees is zero. This is why the string tension on a ball swung in a circle does no work.
What are the units of work?
The SI unit is the joule (J), the same as for energy, because work is energy transferred. One joule is one newton acting over one metre.
References
A quick note on where the physics comes from. Work as Fd cosθ, and the work-energy theorem linking work to the change in kinetic energy, are standard mechanics, set out in OpenStax's University Physics and in Georgia State University's HyperPhysics. The joule and the other SI units follow the US National Institute of Standards and Technology.
- OpenStax, University Physics Volume 1, Section 7.1, Work. https://openstax.org/books/university-physics-volume-1/pages/7-1-work
- HyperPhysics, Georgia State University, Work. http://hyperphysics.phy-astr.gsu.edu/hbase/work.html
- National Institute of Standards and Technology (NIST), Special Publication 811, Guide for the Use of the International System of Units (SI). https://www.nist.gov/pml/special-publication-811
Bibek Lal Karna is a PhD student and graduate teaching assistant at the University of Mississippi, with deep interests in theoretical and gravitational physics. He is also the founder of NRCC and is strongly engaged in scientific teaching and communication. At Eon Tools, he reviews physics tools.