The Scientific Definition of Work

In everyday life, we often describe many activities as “work”—such as writing an exam, holding a heavy object, or walking with groceries. But in physics, the word work has a more precise and limited meaning.

In physics, work is done only when a force causes displacement in the direction of that force.

If there is no movement, or if the movement is perpendicular to the applied force, then no work (in the scientific sense) is done on the object.

The Work Formula

The formal definition of work involves both force and displacement, as well as the angle between them. When a constant force [latex]\mathbf{F}[/latex] is applied and an object is displaced by a distance [latex]\mathbf{d}[/latex], the work done is:

[latex]W = |\mathbf{F}| \cdot |\mathbf{d}| \cdot \cos(\theta)[/latex]

or simply:

[latex]W = Fd\cos\theta[/latex]

Where:

• [latex]W[/latex] is the work done by the force (in joules, J),

• [latex]F[/latex] is the magnitude of the force (in newtons, N),

• [latex]d[/latex] is the displacement of the object (in meters, m),

• [latex]\theta[/latex] is the angle between the force vector and the displacement vector.

Interpreting the Equation

The value of [latex]\cos(\theta)[/latex] determines whether the work is:

• Positive: when the force and displacement are in the same direction ([latex]\theta = 0^\circ[/latex]),

• Negative: when the force and displacement are in opposite directions ([latex]\theta = 180^\circ[/latex]),

• Zero: when the force is perpendicular to displacement ([latex]\theta = 90^\circ[/latex]).

Real-Life Examples of Work

Let’s interpret a few common scenarios using the scientific definition of work:

Figure 42.1(a) – Pushing a lawn mower:
The force [latex]\mathbf{F}[/latex] is applied at an angle [latex]\theta[/latex] to the horizontal. Only the horizontal component of the force contributes to the work done:
[latex]W = Fd\cos\theta[/latex]

Figure 42.1(b)  – Holding a briefcase stationary:
There is a force (upward) but no displacement. Therefore,
[latex]W = 0[/latex] — no work is done on the briefcase.

Figure 42.1(c)  – Carrying a briefcase horizontally at constant speed:
The upward force from your arm is perpendicular to the forward displacement, so:
[latex]\cos(90^\circ) = 0[/latex]
Again, [latex]W = 0[/latex] — no energy is transferred to the briefcase.

Figure 42.1(d) – Carrying a briefcase upstairs:
The upward displacement is in the same direction as the force applied. Here, work is positive and energy is transferred to the briefcase–Earth system.

Figure 42.1(e)  – Lowering a briefcase into a generator:
Here, the generator applies an upward force while the displacement is downward. The angle between force and displacement is [latex]180^\circ[/latex], so:
[latex]\cos(180^\circ) = -1[/latex]
This results in negative work, meaning energy is removed from the briefcase and transferred to the generator.

Units of Work

Since work is the product of force and displacement, its SI unit is:

[latex]1~\text{joule} = 1~\text{newton} \cdot \text{meter} = 1~\text{J} = 1~\text{kg} \cdot \text{m}^2/\text{s}^2[/latex]

• 1 joule is a small amount of energy; for example, lifting a 100-gram apple about 1 meter upward requires roughly 1 joule of work.