Formula & Calculator
Work Done by a Force
Calculates the mechanical work done by a constant force acting on an object as it moves, accounting for the angle between force and displacement.
Interpretation
Work done by a force: W = F·d·cosθ, where F is force, d is displacement, θ is angle between them. Work is energy transferred. Example: 10 N force at 60° to displacement of 5 m → W = 10×5×0.5 = 25 J.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| W | Work done | J |
| F | Applied force magnitude | N |
| d | Displacement of the object | m |
| theta | Angle between the force and displacement direction | degrees |
What it means
Work is defined as the energy transferred to or from an object by a force acting over a displacement. The formula W = F d cosθ accounts for the component of force in the direction of motion. If the force is parallel to displacement, W = Fd; if perpendicular, work is zero. Work is a scalar quantity measured in joules. The work‑energy theorem states that the net work done on an object equals its change in kinetic energy. Work is a central concept in mechanics and thermodynamics, used in analysing machines, engines, and energy systems. In everyday life, work is done when lifting, pushing, or pulling. Understanding work is essential for energy analysis and efficiency calculations.
Worked example
Work Done – Two Examples
Real‑World| Parameter | Value |
|---|---|
| F | 100 N |
| d | 5 m |
| θ | 0° |
| Parameter | Value |
|---|---|
| F | 50 N |
| d | 10 m |
| θ | 60° |
Common mistakes
- Force F and displacement d: Both are vectors – the angle θ is between their directions.
- Work W: Scalar – can be positive, negative, or zero.
- Units: F in N, d in m → J (joules).
- Constant force: This formula assumes constant force; for variable force, use integral ∫F·dr.
- Work done by a force: Only the component of force along displacement does work – perpendicular component does zero work.
Applications
Work done by a force, W = F·d·cosθ, is the energy transferred by a force acting over a distance. It is fundamental to all energy analyses in engineering and physics. Engineers use this formula to calculate the work required to move loads, to design lifting equipment, and to evaluate motor and engine output. In thermodynamics, work is a key term in energy balances. In civil engineering, it is used in the design of dams and excavation. The formula also appears in biomechanics to assess muscular work. By understanding work, professionals can quantify energy consumption, size actuators and motors, and optimise mechanical systems for efficiency and performance.
- Design of lifting and conveying equipment
- Motor and engine power calculations
- Energy balance in thermodynamic cycles
- Excavation and earthmoving work estimation
- Biomechanical analysis of human movement
Frequently Asked Questions
The work done by a constant force is W = F·d·cosθ, where F is the magnitude of the force, d is the magnitude of the displacement, and θ is the angle between the force and displacement vectors.
Forgetting the cosine term. Work is only done by the component of the force in the direction of displacement. If the force is perpendicular to displacement (θ=90°), work is zero.
The SI unit is the joule (J) = N·m = kg·m²/s².
- Positive: force and displacement are in the same direction (θ < 90°), increasing kinetic energy.
- Negative: force and displacement are opposite (θ > 90°), decreasing kinetic energy (e.g., friction).
- Zero: θ = 90° (e.g., centripetal force in uniform circular motion).
The work‑energy theorem states that the net work done on an object equals its change in kinetic energy: W_net = ΔKE. This is a powerful tool for solving problems without knowing acceleration.
If the force varies with position, use integration: W = ∫F(x)·dx (or the appropriate vector dot product). For example, the work done by a spring is ½kx².
Work is force times displacement (linear); torque is force times lever arm (rotational). Work changes translational kinetic energy; torque changes rotational kinetic energy.
Work is the mechanism by which energy is transferred. In a closed system, work done by conservative forces (like gravity) is equal to the negative change in potential energy. This leads to conservation of mechanical energy.