Only the part of a force pointing along the movement does any work, and this is how much. Give it a force, a distance and the angle between them for the answer in joules, kJ, MJ, kWh and kcal.
Work Calculator - W = Fd cos, Joules from Force
Only the part of a force pointing along the movement does any work, and this is how much. Give it a force, a distance and the angle between them for the answer in joules, kJ, MJ, kWh and kcal.
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What counts as work, and what only feels like it
Carry a heavy backpack a kilometer along a level path and you have done no work at all in the sense this calculator means. Push a shopping cart 30 m with 40 N and you have done 1.20 kJ. The difference is not effort, it is direction, and it is the reason the formula for work carries a cosine that the formula for energy does not. Enter a force, a distance and the angle between them and the calculator returns the work in joules, kilojoules, megajoules, kilowatt hours and kilocalories. Enter a work figure instead and it returns the force or the distance that produced it.
The angle is the part that catches people
Work is W = F × d × cos α, where alpha is the angle between the direction the force points and the direction the object actually moves. Only the part of the force lying along the path does anything. The rest is real, it is measurable, and it is doing nothing whatsoever to the energy of the object.
Pull a suitcase by a handle held at 30° with 200 N and the force splits into 173.205 N dragging it forward and 100 N lifting it. Those two rebuild the original 200 N as the sides of a right-angled triangle rather than as a sum, which is why they do not add to 273. Over 10 m the forward component does 1.73 kJ and the upward component does exactly nothing.
The table below holds that same force and that same distance fixed and moves only the angle, which is the fastest way to see how much a badly aimed pull costs. The calculator prints this table under every result using your own two numbers, with your own angle highlighted.
| Angle | cos | Work by 200 N over 10 m | Share of a straight pull |
|---|---|---|---|
| 0° | 1.0000 | 2.00 kJ | 100.0% |
| 15° | 0.9659 | 1.93 kJ | 96.6% |
| 30° | 0.8660 | 1.73 kJ | 86.6% |
| 45° | 0.7071 | 1.41 kJ | 70.7% |
| 60° | 0.5000 | 1.00 kJ | 50.0% |
| 75° | 0.2588 | 517.6 J | 25.9% |
| 90° | 0.0000 | 0 J | 0.0% |
| 120° | -0.5000 | -1,000.0 J | -50.0% |
| 150° | -0.8660 | -1.73 kJ | -86.6% |
| 180° | -1.0000 | -2.00 kJ | -100.0% |
Notice how forgiving the top of that table is. At 15° you still get 96.6% of a straight pull, and even at 30° you keep 86.6%. The collapse happens late: between 60° and 90° the work falls from half to nothing. A handle angled slightly wrong is barely worth correcting; a handle angled badly wrong is worth almost nothing.
Which two numbers you already have
- What to solve for - the work, the force or the distance. The form shows nothing else until this is chosen, and no option is preselected.
- Force unit - newtons, kilonewtons, kilogram-force or pounds-force. One kgf is 9.80665 N and one lbf is 4.4482216152605 N, both exact, so a figure taken off an American spec sheet and the same figure in newtons give one identical answer.
- Distance unit - meters, centimeters, kilometers or feet, at 0.3048 m to the foot. This is the distance the object moves, not the length of the rope or the reach of your arm.
- Force - always entered as a positive number. Which way it points is carried by the angle, never by a minus sign, because a force entered as negative and an angle of 180 degrees would cancel each other into a wrong answer.
- Distance - how far the object actually travels while that force is acting. If the force stops halfway, only the first half counts.
- Angle - optional, from 0 to 180 degrees, and decimals are allowed. Blank means zero, a force straight along the motion. Ninety means perpendicular and no work. Anything above ninety means the force is fighting the movement.
- Read the answer. The headline gives the quantity you asked for, the line below restates it in five units, the note beneath splits the force into the part that works and the part that does not, and the table sweeps your own numbers across ten angles.
Seven jobs, worked all the way through
Each row carries the three numbers that made it, so any of them can be checked with a calculator and a cosine key. Two of them are worth reading twice: the backpack, where a very large force over a very long distance produces nothing, and the crate, where the work comes out below zero.
| Task | Force | Distance | Angle | Work |
|---|---|---|---|---|
| Pushing a shopping cart down an aisle | 40 N | 30 m | 0° | 1.20 kJ |
| Pulling a suitcase by its handle | 25 N | 200 m | 40° | 3.83 kJ |
| Lifting a 20 kg case into an overhead bin | 196.133 N | 1.2 m | 0° | 235.4 J |
| Carrying a backpack along a level corridor | 150 N | 1,000 m | 90° | 0 J |
| An adult climbing one flight of stairs | 735.499 N | 3.3 m | 0° | 2.43 kJ |
| Friction on a crate dragged across concrete | 150 N | 8 m | 180° | -1.20 kJ |
| A tow rope pulling a car onto a flatbed | 3,000 N | 500 m | 20° | 1.41 MJ |
The stairs row is the same job the potential energy calculator prices at 2.43 kJ for a 75 kg adult climbing 3.3 m, reached from the other direction. There the input is a mass and a height; here it is the weight of that mass, 735.499 N, dragged through the same 3.3 m. The two agree to the joule because they are the same statement, and that agreement is the whole content of the phrase work done against gravity.
Positive, negative and zero, and why all three are real
Positive work adds energy to the object. The tow rope makes the car move; the cart ends up rolling. Negative work takes energy away, and every braking force in the world does it: friction on the crate removes 1.20 kJ over its 8 m, which is why the crate stops and why the floor beneath it warms up. A minus sign in the answer is not an error message.
Zero work is the one that provokes arguments. The backpack row above has a force of 150 N acting over a full kilometer and still reports 0 J, because holding something up while walking on the level means the force and the movement are at right angles the whole way. The objection is always the same, and it is a fair one: carrying a backpack for a kilometer is exhausting. It is, and the energy is entirely real, but it is spent inside your muscles holding a static contraction, not transferred to the backpack. The backpack finishes the walk with exactly the energy it started with.
Five ways to end up with a smaller number than you expected
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