- W = F·d·cosθ (constant force)
- W = ∫ F(x) dx (variable force)
- W_g = ±mgh (gravity)
- W_f = −μNd (friction)
- W_on_spring = ½k(x2²−x1²)
- W_net = ΔK = ½m(v2²−v1²)
Work Calculator Guide
1) What “work” means in physics
Work measures energy transfer when a force causes displacement. The calculator follows the standard definition: the dot product of force and displacement. If the force points with motion, work is positive. If it opposes motion, work is negative. Units are joules (J), where 1 J = 1 N·m.
2) Constant-force work with angles
For constant forces, the core model is W = F·d·cosθ. Enter the force magnitude, displacement, and the angle between them. The tool also reports the parallel component F∥ = F·cosθ, because only that component contributes to work. Angles are in degrees in these constant-force modes for convenience.
3) Multiple forces and net work
Real systems often have several forces at once. In “Multiple forces,” each force can have its own angle, and the calculator sums individual works. If you already know the net force along the motion line, you can use the simpler net-force mode, W = Fnet·d, to compute total work directly.
4) Gravity and elevation change
For vertical motion, gravity work depends on direction. Lifting an object upward gives Wg = −mgh, and lowering gives Wg = +mgh. This calculator uses g = 9.80665 m/s². It also shows approximate applied work for slow lifting/lowering, which is the opposite sign of gravity work in quasi-static cases.
5) Friction: typical inputs and signs
Friction work is always negative for motion along the surface because friction opposes motion: Wf = −μNd. On a horizontal surface, N = mg. On an incline, N = mg·cosα. Typical kinetic friction coefficients range from about 0.02 (polished/rolling) up to 0.8+ (rubber on rough surfaces), depending on materials.
6) Inclined-plane toolbox breakdown
The incline toolbox separates work by gravity, normal, friction, and applied force. The normal force is perpendicular to motion, so its work is approximately zero. Gravity work changes sign when moving up versus down the slope. Adding applied force lets you estimate how much work a motor or person supplies over a distance.
7) Spring work and energy storage
For springs, the tool uses W_on = ½k(x2² − x1²) to compute work done on the spring, which equals the change in elastic potential energy. It also reports work done by the spring force as the negative of that value. Use consistent meters for x and N/m for k to keep units correct.
8) Variable-force integration and accuracy
When force varies with position, work is the area under the F(x) curve: W = ∫F(x)dx. This calculator evaluates the integral numerically using Simpson’s rule with an even number of slices n. Increase n for better accuracy. Trigonometric functions in F(x) use radians, which is standard in most physics formulas.
FAQs
1) Why can work be negative?
Work is negative when the force component along motion points opposite the displacement. This usually means energy is being removed from the moving object, such as friction slowing it down.
2) What angle should I enter for θ?
Enter the angle between the force vector and the displacement direction. For pushing straight forward, θ = 0°. For pulling backward against motion, θ = 180°.
3) Does the normal force do work on an incline?
In ideal sliding motion along the plane, the normal force is perpendicular to displacement, so its work is approximately zero. Small deviations can occur if the path is not perfectly along the surface.
4) What value of g does the calculator use?
The calculator uses g = 9.80665 m/s², a standard reference gravitational acceleration. If you need a local value, you can adjust your inputs accordingly in custom scenarios.
5) My F(x) uses sin(x). Are angles degrees?
In the variable-force expression, trigonometric functions use radians. If your model uses degrees, convert first (for example, use x·π/180 inside sin).
6) How accurate is the variable-force integral?
Accuracy depends on how smooth F(x) is and the slice count n. Simpson’s rule converges quickly for smooth functions. Increase n if the curve changes rapidly or has sharp bends.
7) Can I export multiple history rows?
The export buttons download the latest result shown at the top. Use the history list to verify older runs, then re-calculate that case to export it again.