Work Done Calculator
Enter the force, displacement and angle to compute the work W=F·d·cosθ, quantifying the energy transferred by a force. Pushing 50 N over 3 m (same direction) → W=150 J.
Input Data
Results
At a glance:Work done W=F·d·cosθ is the physics quantity describing the energy transferred by a force as it moves an object — the bridge between force and energy. Colloquially 'work' is vague, but in physics work is precisely defined: a force does work only when it produces a displacement along its direction. F is the force magnitude (N); d the displacement (m); θ the angle between force and displacement. Together W=F·d·cosθ, in joules (J=1 N·m). The cosθ factor keeps only the component of force parallel to the displacement (F·cosθ), since only that component drives the motion. When force and displacement are parallel (θ=0°, cos0°=1) work is maximal W=F·d; at an angle it is scaled by cosθ; when force is perpendicular to displacement (θ=90°, cos90°=0) it does no work — that is why carrying a weight horizontally, the upward supporting force does no work (vertical force, horizontal motion), and why centripetal force in uniform circular motion does no work (always radial vs tangential motion). For θ>90° (friction, drag, opposing the motion) cosθ is negative, so work is negative — the force removes energy. Example: a 50 N horizontal push over 3 m (θ=0°) → W=50×3×1=150 J; at 60° → W=50×3×0.5=75 J, half as much. Work and energy are inseparable: the work-energy theorem says the net work of all forces equals the change in kinetic energy (W_net=ΔKE=½mv²−½mv₀²); gravity's work equals the loss of gravitational potential energy, making work the core tool for analyzing energy conversion.
Formula
Work: W = F·d·cosθ (force × displacement × cosine of angle).
Maximum when parallel (θ=0°): W = F·d.
Work-energy theorem: net work = change in kinetic energy W = ΔKE.
$$W = F d \cos\theta$$$$W_{\text{net}} = \Delta KE = \tfrac{1}{2}mv^2 - \tfrac{1}{2}mv_0^2$$How to Use
- Enter the applied force F (N).
- Enter the displacement d (m).
- Enter the angle θ between force and displacement (°; 0 if same direction).
- The tool computes the work W = F·d·cosθ (J).
Case Studies
Pushing a box horizontally
F=50 N, d=3 m, θ=0° → W=50×3×cos0°=150 J.
At θ=60°, W=50×3×cos60°=75 J (only the horizontal component works).
The angle directly halves the work done.
Lifting vs carrying
Lifting straight up: force and displacement both vertical, θ=0°, work = mgh.
Carrying horizontally at constant height: supporting force is vertical, motion horizontal, θ=90° → zero work by the support.
This distinction clears up many everyday 'why no work?' puzzles.
FAQ
What is work in physics?
W=F·d·cosθ — the energy a force transfers while moving an object. Only the force component along the displacement contributes; the angle θ decides how much.
Why is no work done when θ=90°?
Because cos90°=0. A force perpendicular to the displacement (e.g. the upward support while walking horizontally, or centripetal force in circular motion) transfers no energy along the motion.
What is negative work?
When θ>90° the force opposes the motion (friction, drag), cosθ<0 and work is negative — the force removes energy from the object.
How is work linked to energy?
The work-energy theorem: net work equals the change in kinetic energy. Gravity's work equals the drop in potential energy, so work is the central bookkeeping quantity for energy conversion.
Is work the same as torque (N·m)?
Same dimension (N·m) but different meaning: work is a scalar energy transfer (J); torque is a rotational influence (N·m) that need not involve motion. Do not confuse them.
Related Tools
References
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.