Work Calculator

Calculate work done, applied force, or displacement using W = F·d·cos(θ). Includes angle adjustment for forces not parallel to motion.

🔧 Energy📐 W = Fd·cosθ⚙️ Mechanics
Force (F)
Unit
Distance (d)
Unit
Angle θ (degrees, 0 = parallel)
⚠️ Please enter valid positive numbers. Check angle is 0–180°.

What Is Work in Physics?

In physics, work has a precise meaning distinct from everyday usage. Work is done when a force causes displacement in the direction of that force. The formula is W = F·d·cos(θ), where F is the magnitude of the applied force (N), d is the displacement (m), and θ is the angle between the force vector and the direction of motion. The SI unit is the joule (J) — equal to one newton-meter (N·m).

The cosine factor is critical and often overlooked. Only the component of force in the direction of motion does work. Push a box at 30° to the floor: only F·cos(30°) contributes to horizontal displacement. At θ = 90° (force perpendicular to motion), cos(90°) = 0 — no work is done. This is why a person carrying a heavy box horizontally across a room does zero work against gravity: the carrying force is vertical while displacement is horizontal.

Work can be negative. Friction opposes motion (θ = 180°, cos = −1), so friction always does negative work, removing energy from the system. Gravity does negative work when you lift an object but positive work when it falls.

The work-energy theorem states that the net work done on an object equals its change in kinetic energy: W_net = ΔKE = ½mv_f² − ½mv_i². This powerful principle allows you to find the velocity of an object after a force acts over a distance, even without knowing time. And the work-power relationship connects work to the rate at which it is done: P = W/t (power equals work per unit time).

Formula Reference Table

Solve ForFormulaNotes
Work (W)W = F · d · cos θθ = angle between force and displacement
Force (F)F = W / (d · cos θ)Requires cos θ ≠ 0 (θ ≠ 90°)
Distance (d)d = W / (F · cos θ)Displacement in direction of motion
Work-Energy TheoremW_net = ΔKE = ½mv_f² − ½mv_i²Net work = change in KE
Work by gravityW_gravity = mghPositive when falling (h > 0)
Work by springW_spring = ½kx²Energy stored in spring

3 Worked Examples

Example 1
Pushing a Box — Angled Force

A person pushes a 20 kg box with 150 N at 25° below horizontal, moving it 8 m. How much work is done?

  • Effective horizontal force: F_x = 150 × cos(25°) = 150 × 0.906 = 135.9 N
  • Work: W = F · d · cos θ = 150 × 8 × cos(25°) = 1,087 J
  • Note: The vertical component (150 × sin25° = 63.4 N) presses into the floor — does no work on horizontal motion
✓ Work done = 1,087 J (1.087 kJ)
Example 2
Work-Energy Theorem — Find Final Velocity

A 5 kg object starts at rest. A net force of 40 N acts over 3 m. What is the final speed?

  • Net work: W = F × d = 40 × 3 = 120 J (θ = 0°, force parallel to motion)
  • Work-energy theorem: W = ½mv_f² − 0 → 120 = ½ × 5 × v_f²
  • Solve: v_f² = 240/5 = 48 → v_f = √48 = 6.93 m/s
✓ Final speed = 6.93 m/s
Example 3
Zero Work — Carrying vs. Lifting

A 70 kg person carries a 15 kg box 50 m horizontally. How much work is done against gravity?

  • Gravity force: F_g = mg = 15 × 9.8 = 147 N (downward)
  • Displacement: 50 m horizontal (rightward)
  • Angle between force and displacement: θ = 90°, cos(90°) = 0
  • Work by gravity: W = 147 × 50 × 0 = 0 J
  • The person's muscles do work against their own body (vertical oscillations), but gravity does zero work
✓ Work done against gravity = 0 J (horizontal carry, force ⊥ displacement)

Real-World Applications

🏋️
Weightlifting
Lifting a 100 kg barbell 2 m does W = 100 × 9.8 × 2 = 1,960 J of work against gravity. Lowering it does −1,960 J (negative work by you, positive by gravity). Machines calculate rep work for calorie estimation.
🏗️
Construction Cranes
Crane operators calculate work done lifting materials (W = mgh) to size motors and estimate fuel consumption. The efficiency of the hoist determines how much electrical energy (W_in) converts to useful work (W_out = mgh).
⚡
Electrical Energy
Electrical work W = QV (charge × voltage) is directly analogous to mechanical work W = Fd. Moving charge through a potential difference does work that converts to heat, light, or mechanical energy. The joule is the universal unit.
🔩
Mechanical Engineering
In machine design, work analysis identifies energy losses. An inefficient gear train may input W_in = 500 J but output only W_out = 380 J — 120 J lost to friction. Efficiency = W_out/W_in = 76%.
🌊
Wave Energy
Ocean wave energy converters harness the work done by wave forces on floating structures. The power output depends on force amplitude and displacement: P = W/t = F × v_piston, where v_piston is the oscillation velocity.

Common Mistakes to Avoid

⚠️
Forgetting the angle (using W = F×d instead of W = Fd·cosθ)

If you push at 30° and use W = Fd, you overestimate work by 1/cos(30°) ≈ 15%. Always resolve the force into the component parallel to displacement, or use the cosine factor explicitly.

⚠️
Confusing work with force or energy

Work (J) is energy transferred. Force (N) is what causes acceleration. A force can exist without doing work (a wall holding up a shelf exerts force but does zero work since d = 0). Work requires both force and displacement in the same direction.

⚠️
Thinking work depends on time

Work W = Fd is independent of how long the force acts. Whether you push a box 10 m in 1 second or 10 minutes, the work done is identical. Time matters for power (P = W/t), not work.

⚠️
Ignoring the sign of work

Work can be negative (friction, gravity during lift). When using the work-energy theorem, you must sum all work including negative contributions. Friction's negative work reduces final kinetic energy.

⚠️
Using distance instead of displacement

Work uses displacement (straight-line distance from start to finish), not total path length. A person who walks 100 m along a curved path and ends 20 m from the start has displacement = 20 m (for work against friction, use path length, but for gravity, use vertical height change).

Frequently Asked Questions

What is the difference between work and energy? ▾
Energy is the capacity to do work; work is the process of transferring energy. Both are measured in joules. When you do 500 J of work on an object, 500 J of energy is transferred to it (as kinetic energy, potential energy, or heat). Energy is a state quantity (what an object has); work is a process quantity (what happens during a transfer).
Why does a person carrying a box horizontally feel tired if no work is done on the box? ▾
While no net work is done against gravity on horizontal displacement, the human body is not a simple force machine. Muscles contract repeatedly (sarcomeres cycle) to maintain the static force of holding the box, consuming ATP energy each cycle. Also, the body naturally bobs up and down while walking, doing small amounts of positive and negative work each step. Fatigue reflects physiological energy expenditure, not the simple physics definition of 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 = ½mv_f² − ½mv_i². It unifies force and energy analysis: instead of using F = ma and kinematic equations to find velocity, you can directly calculate net work done and derive the final speed. It is especially powerful for problems with variable forces (like spring forces) where F = ma is harder to integrate.
Can work be done without motion? ▾
No — by definition, W = Fd·cosθ requires d > 0. A person pushing on a stationary wall exerts force but does zero work (d = 0). Holding a weight overhead requires force but does no work against gravity. This is where physics diverges from everyday language: in common speech, "hard work" can mean straining without moving; in physics, it means energy transferred through displacement.
How is work different from torque? ▾
Torque (τ = r × F, units N·m) and work (W = F·d, units J = N·m) share the same dimensional units but are physically different. Torque is a rotational force (causes angular acceleration); work is energy transferred. In rotational systems, work done = τ × θ (torque × angle in radians), so the angular version of W = Fd is W = τθ.
What is negative work? ▾
Negative work occurs when the force component opposes displacement (θ between 90° and 180°, so cosθ < 0). Friction always does negative work — it removes energy from moving objects. Gravity does negative work when you lift an object (force down, displacement up). Negative work reduces the kinetic energy of the object; it is energy removed from the object, not added.
How many joules in a calorie? ▾
1 thermochemical calorie = 4.184 J. The "Calorie" on food labels is actually a kilocalorie (kcal) = 4,184 J. So a 500 Calorie meal contains 2,092,000 J = 2.09 MJ of chemical energy. In comparison, lifting 100 kg by 2 m takes about 1,960 J — you'd need to do nearly 1,068 such lifts to burn off one food Calorie.
How does work relate to potential energy? ▾
Work done against a conservative force (gravity, spring) is stored as potential energy. Lift a 10 kg mass 5 m: you do W = mgh = 490 J of work against gravity, and the mass gains 490 J of gravitational PE. When released, gravity does 490 J of work on the mass, converting PE back to KE. Non-conservative forces (friction) convert work to heat — permanently lost from mechanical energy.

Related Physics Calculators

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Formula Explorer connections

Interpretation: This relationship connects motion, force, momentum, work or energy in a mechanical system. Assumption: Choose a consistent reference direction and unit system. The model may assume constant acceleration, rigid bodies, negligible losses or an isolated system.

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