Work by Constant Force Along Motion
In physics, work has a precise meaning that differs from everyday usage. Work is done on an object when a force causes it to move through a displacement. The simplest case — and the foundation for everything else in energy physics — is when a constant force acts exactly along the line of motion. Imagine pushing a heavy box along a flat floor: if you push horizontally and the box slides horizontally, your force and the displacement point in the same direction. This is the parallel case, and it represents the maximum possible work a given force can do over a given distance. The antiparallel case is equally important. Friction, for example, acts opposite to the direction of motion. When a braking force slows a car, the force points backwards while the displacement points forwards — they are antiparallel. In this situation the work done by friction is negative, meaning energy is being removed from the car's motion. Understanding the sign of work is crucial: positive work adds energy to an object, whilst negative work removes it. The net work done by all forces on an object equals the change in its kinetic energy — a relationship known as the work–energy theorem. This article focuses exclusively on the case where the angle between force and displacement is either 0° (parallel) or 180° (antiparallel). This keeps the mathematics clean and builds the intuition needed before tackling forces at arbitrary angles. Mastering this case first makes the general formula — which involves the cosine of the angle — far more understandable when you encounter it later.
- Define the Scenario
- Identify Whether the Force is Parallel or Antiparallel
- Apply the Sign Convention
- Interpret the Result
- Connect to the Work–Energy Theorem
Define the Scenario
Choose a clear positive direction for displacement. Displacement is a vector pointing from the object's starting position to its finishing position. The magnitude of displacement is the straight-line distance travelled, denoted s (in metres). The force F is constant — it does not change in magnitude or direction throughout the motion.
Identify Whether the Force is Parallel or Antiparallel
If the force points in the same direction as the displacement (angle = 0°), the force is parallel. Examples: a engine thrust propelling a rocket upwards; a hand pushing a trolley forward. If the force points in the opposite direction to displacement (angle = 180°), the force is antiparallel. Examples: air resistance opposing a moving cyclist; gravity acting downward while a ball falls upward after being thrown.
Apply the Sign Convention
For a parallel force (force and displacement in the same direction): W = +Fs. The work is positive, meaning the force transfers energy into the object. For an antiparallel force (force and displacement in opposite directions): W = −Fs. The work is negative, meaning the force removes energy from the object. Both cases are captured by the general formula W = Fs cos θ, where θ = 0° gives cos 0° = +1 and θ = 180° gives cos 180° = −1.
Interpret the Result
Work is a scalar quantity measured in joules (J). One joule equals one newton-metre (1 J = 1 N·m). The sign tells you the direction of energy transfer: positive work means energy flows from the force's source into the object; negative work means energy flows out of the object. If multiple forces act simultaneously, you can calculate the work done by each force separately, then add them (with their signs) to find the total (net) work done on the object.
Connect to the Work–Energy Theorem
The net work done on an object equals its change in kinetic energy: W_net = ΔKE = ½mv² − ½mu², where u is initial speed and v is final speed. This is why positive work speeds an object up and negative work slows it down. For example, a 5 N braking force acting over 4 m does W = −5 × 4 = −20 J of work, removing 20 J from the object's kinetic energy.
The Key Equation
- W — Work (joule (J)): The scalar quantity representing the energy transferred by the force. Positive if energy is added to the object, negative if energy is removed.
- F — Force magnitude (newton (N)): The magnitude of the constant force acting on the object. Always taken as a positive value; direction is handled by the cosine term.
- s — Displacement magnitude (metre (m)): The straight-line distance between the object's initial and final positions. Always positive.
- \theta — Angle between force and displacement (degrees or radians (dimensionless)): The angle measured between the direction of the force vector and the direction of the displacement vector. For parallel: 0°; for antiparallel: 180°.
Worked Example
Problem. A removal worker pushes a wardrobe along a straight corridor. She applies a horizontal force of 120 N in the direction of motion for a distance of 3.5 m. A frictional force of 45 N acts on the wardrobe, opposing its motion throughout. (a) Calculate the work done by the applied force. (b) Calculate the work done by friction. (c) Find the net work done on the wardrobe.
Answer. Work by applied force = +420 J; work by friction = −157.5 J; net work = +262.5 J
Origins · 1 minGaspard-Gustave de Coriolis and how this was discovered
Coriolis formally introduced the modern scientific definition of 'work' (travail in French) in his 1829 treatise 'Du Calcul de l'Effet des Machines'. He defined work as force multiplied by displacement in the direction of the force — exactly the formula W = Fs used today. Before Coriolis, the concept existed informally but lacked a consistent name and definition. His formulation unified the study of machines and energy transfer, laying groundwork for classical mechanics and engineering thermodynamics. Coriolis is most famous today for the Coriolis effect — the deflection of moving objects caused by Earth's rotation — yet he discovered this whilst studying mechanics of rotating machinery, not meteorology. The connection to weather and ocean currents was made by later scientists.
Related Concepts
- Work by a Force at an Angle — The natural extension of this concept: when force is not parallel or antiparallel to displacement, the cosine component of the angle must be included, generalising W = Fs cos θ.
- Kinetic Energy — The work–energy theorem directly links net work done to the change in kinetic energy (W_net = ΔKE), making kinetic energy the key quantity affected by the work calculated here.
- Power and Rate of Work — Power is defined as work done per unit time (P = W/t). Understanding how to calculate work correctly is a prerequisite for determining power output of engines and motors.
- Gravitational Potential Energy — Gravity is a constant force near Earth's surface. Work done against gravity (antiparallel case) equals the gain in gravitational potential energy, connecting this concept directly to energy storage.