Work Formula: W = Fd cos θ, Work Done Formula With Examples

The work formula is $W = Fd\cos\theta$ : work done equals the force, times the displacement, times the cosine of the angle between them. Push a crate with 50 N across 4 m of floor in the direction it moves and you do 200 J of work on it. Work is measured in joules, and one joule is the work done when a force of one newton moves its point of application one metre along its own line.

Indian textbooks usually call this the work done formula and write it in vector form as $W = \vec{F} \cdot \vec{s}$ or $W = \vec{F} \cdot \vec{r}$ , the dot product of force and displacement. US books often use $W = Fd\cos\theta$ with d for displacement. The physics is the same: only the part of the force along the motion does any work. The calculator on this page solves the work formula for W, F, d or the angle θ.

Quick facts

Formula
$W = Fd\cos\theta$
Vector form
$W = \vec{F} \cdot \vec{s}$
SI unit
joule (J), where 1 J = 1 N m
Dimensional formula
[M¹ L² T⁻²]
Type
Scalar, but it can be positive, negative or zero
Variable force
$W = \int F\,dx$ , the area under a force against displacement graph
Other units
1 erg = 10⁻⁷ J; 1 kWh = 3.6 × 10⁶ J

Formula of work done, symbol by symbol

Every letter in W = Fd cos θ
Letter	What it stands for	Unit
$W$	Work done by the force	J
$F$	Size of the constant force	N
$d$ or $s$	Size of the displacement	m
$\theta$	Angle between the force and the displacement	degree or rad

The cos θ factor picks out the part of the force along the displacement. When the force points the same way as the motion, θ = 0 and cos θ = 1, so $W = Fd$ . That simple product is the formula for work most Class 9 problems use. When the force is at an angle, only $F\cos\theta$ pushes the object along; the other part, $F\sin\theta$ , does no work.

Positive, negative and zero work

How the angle decides the sign
Angle θ	cos θ	Work	Example
0°	1	Positive, maximum	Engine force on a car moving forward
Between 0° and 90°	Positive	Positive	Pulling a suitcase by an angled handle
90°	0	Zero	Gravity on a ball rolling on a level floor
Between 90° and 180°	Negative	Negative	Gravity on a ball thrown upward at an angle, while it climbs
180°	−1	Negative, −Fd	Friction on a sliding box

Zero work catches many students. A waiter carrying a tray across a room at steady height does no work on the tray against gravity, because the upward force is at 90° to the horizontal displacement. A planet in a circular orbit has zero work done on it by gravity for the same reason. Friction almost always does negative work, a case worked through on the work done by friction page.

Worked examples with the formula for work done

Example 1: a straight push

A shopper pushes a trolley with a horizontal force of 50 N for 4.0 m. Find the work done.

$W = Fd\cos 0^\circ = 50 \times 4.0 \times 1 = 200\ \text{J}$

Example 2: pulling at an angle

A traveller drags a suitcase 25 m along a level platform, pulling the handle with 40 N at 60° above the horizontal. Find the work done by the pull.

$W = 40 \times 25 \times \cos 60^\circ = 1000 \times 0.5 = 500\ \text{J}$

Only half of the pull acts along the platform, so only half of 1000 J is done.

Example 3: lifting against gravity

A student lifts a 2.0 kg book from the floor to a shelf 1.5 m up at steady speed. How much work does she do?

Her upward force matches the book's weight of 19.6 N (that is 2.0 kg times 9.8 m/s²) and points along the displacement.

$W = mgh = 19.6 \times 1.5 = 29.4\ \text{J} \approx 29\ \text{J}$

Gravity does −29 J on the book during the same lift.

Example 4: an exam style question with unit changes

A force of 0.50 kN acts on a cart at 37° to its path while the cart moves 300 cm. Find the work in joules and in kilojoules. Take cos 37° = 0.80.

Convert: F = 500 N, d = 3.00 m.

$W = 500 \times 3.00 \times 0.80 = 1200\ \text{J} = 1.2\ \text{kJ}$

Work formula physics students use for a variable force

When the force changes along the way, split the path into tiny steps dx and add up $F\,dx$ for each step:

$W = \int_{x_1}^{x_2} F\,dx$

On a graph of force against displacement, this integral is the area under the line. For a spring obeying $F = kx$ , the graph is a triangle and the work needed to stretch it by x is $\frac{1}{2}kx^2$ , which is the potential energy of a spring. AP Physics 1 and Class 11 questions often give a force graph and ask for the work: count the area, including any part below the axis as negative.

Units, dimensions and what work changes

Since one newton equals one kg m/s², one newton metre comes to kg m²/s², named the joule. Its bracket of base powers, [M¹ L² T⁻²], is identical to energy; the dimensional formula of work page sets out each step. That match is not a coincidence. The work energy theorem says the net work done on an object equals its change in kinetic energy, $W_{\text{net}} = \Delta K$ , as shown on the work energy theorem page.

Torque is also quoted in N m, yet nobody writes a torque in joules. Work multiplies the force by the displacement it shares a direction with; torque multiplies a force by a lever arm drawn perpendicular to it, and nothing need move at all.

Slips that lose marks

Using the angle with the vertical when the formula needs the angle between F and d. Draw both arrows and measure between them.
Calling all work positive. Friction and braking forces do negative work.
Plugging in the total path length for a constant force. A box pushed 3 m out and 3 m back by a constant force has zero net displacement.
Leaving km, cm or kN unconverted. Work in SI units gives joules.
Thinking holding a heavy bag still is work. Your muscles tire, but with no displacement W = 0 in physics.

For the wider chapter, including power and energy, start from the work and energy hub.

Frequently asked questions

What is the formula for work in physics?

Work done is W = Fd cos θ, where F is the force, d the displacement and θ the angle between them. If the force is along the motion, W = Fd.

Which unit is work done measured in?

The joule (J). One joule is one newton of force acting through one metre of displacement, so 1 J = 1 N m.

What is the value of r in the work done formula?

In W = F · r, r is the displacement vector of the point where the force acts, from its starting position to its final position, measured in metres. It is not a radius.

Does work have a direction?

Work is a scalar. It comes from the dot product of two vectors, force and displacement, which gives a number with no direction. It can still be negative.

When is work done zero?

When there is no displacement, or when the force is perpendicular to the displacement so that cos 90° = 0. A stone spun on a string in a level circle is a good case: the string pulls inward while the stone moves sideways.

How are work and energy different?

Energy is the capacity to do work, and work is the transfer of energy by a force through a displacement. Both are measured in joules.