Positive, Negative, and Zero Work
Work is positive when the force has a component in the direction of the displacement (it adds energy to the object), negative when the force has a component opposite to the displacement (it removes energy), and zero when the force is perpendicular to the displacement or there is no displacement. In brief: the sign of W = Fs cos θ follows the sign of cos θ: positive for angles below 90°, zero at 90°, negative above 90°. Positive work increases the energy of the object, and negative work decreases it, as friction and braking forces do.
Several forces usually act on an object, and each does its own amount of work, which can be positive, negative or zero. The total work is their sum. This guide explains the three cases, relates them to energy, compares examples in a table, works through a block-on-a-slope example and corrects common mistakes.
When is work positive?
Work is positive when the force acts at an angle of less than 90° to the displacement (0° ≤ θ < 90°). The force helps the motion and transfers energy to the object. Examples: a person pushes a car forward, gravity acts on a falling apple (force down, displacement down), and the engine’s driving force accelerates a car. The positive work increases the kinetic energy or another store of the object.
When is work negative?
Work is negative when the force acts at an angle of more than 90° to the displacement (90° < θ ≤ 180°). The force opposes the motion, and takes energy away from the object. Examples:
- Friction and air resistance on a moving object: the work is negative, and energy is transferred to the thermal store.
- Braking forces: they reduce the kinetic energy of the car.
- Gravity on a rising ball: the force is down while the displacement is up, so gravity does negative work, and the kinetic energy decreases as the potential energy increases.
- Lowering a weight slowly: the lifting force is up while the displacement is down, so the person does negative work on the weight.
When is work zero?
- No displacement: pushing against a wall, or holding a weight still.
- Force perpendicular to displacement (θ = 90°): the normal force on an object sliding along a horizontal surface, the tension in a pendulum string (always perpendicular to the motion of the bob), the centripetal force on an object in uniform circular motion, and the weight of an object moving horizontally.
- Zero force: an object moving with no force on it (such as a puck gliding on a frictionless surface) has no work done on it.
Positive, negative and zero work compared
| Angle θ | cos θ | Work | Effect on energy | Example |
|---|---|---|---|---|
| 0° | +1 | +Fs | Increases | Pushing a trolley |
| 60° | +0.5 | +0.5Fs | Increases | Pulling a sled by a rope |
| 90° | 0 | 0 | No change | Normal force on a floor |
| 180° | −1 | −Fs | Decreases | Friction |
Worked example: work by each force on a sliding block
A 2.0 kg block slides 3.0 m down a rough slope inclined at 30° to the horizontal. Friction is 4.0 N. Use g = 9.8 m/s2.
- Weight: mg = 19.6 N. Its component along the slope is 19.6 × sin 30° = 9.8 N, in the direction of motion. Work by gravity = +9.8 × 3.0 = +29.4 J (positive).
- Normal force: perpendicular to the slope, so the angle with the displacement is 90°. Work = 0.
- Friction: 4.0 N opposing the motion (θ = 180°). Work = −4.0 × 3.0 = −12 J (negative).
- Total (net) work: 29.4 + 0 − 12 = +17.4 J. By the work-energy theorem, the kinetic energy gained is 17.4 J. If the block starts from rest, ½ × 2.0 × v2 = 17.4, so v = 4.2 m/s.
How was the sign convention established?
The concept of positive and negative work developed together with the analysis of machines in the 19th century. It was formalized in the vector notation, where work is the scalar product of force and displacement vectors, introduced by Gibbs and Heaviside in the 1880s. The sign convention tells whether energy flows into or out of a system.
Which mistakes are common?
Mistake 1: Thinking that negative work means that no work is done
It means that the force takes energy from the object.
Mistake 2: Assuming that friction always does negative work
Friction can do positive work, for example, static friction on the foot of a person walking pushes the person forward, or on a box carried by an accelerating truck. The sign depends on the direction of the displacement of the point where it acts.
Mistake 3: Saying that the normal force never does work
It does no work when it is perpendicular to the displacement, but in an accelerating elevator it does work on a person.
Mistake 4: Adding the magnitudes of work instead of the signed values
The net work is the sum of the signed values.