NDA Physics · Work, Energy and Power

Work — Force Times Displacement Times Cosine

Work is done when a force moves its point of application through a displacement. Only the part of the force along the displacement counts, so W = F d cos θ — which makes work zero when force is perpendicular to motion and negative when it opposes motion.

All Work, Energy and Power notes NDA Physics strategy

Why this matters

This is the foundation of the whole chapter — every energy and power result is built on the definition of work. The NDA tests it almost entirely through the sign and the angle: zero work when force is perpendicular, negative work when force is anti-parallel, and the definition of one joule. 5 PYQs, all EASY or MODERATE. Get the cosine rule and its sign cases right and these are free marks.

Concept 1 of 3

What work means in physics — W = F d cos θ

Intuition

In everyday language, holding a heavy bag is hard work. In physics it is not work at all — nothing moved in the direction of the force. Physics work happens only when a force actually pushes its target through a displacement, and it counts only the part of the force that lies ALONG that displacement.

Definition

Work done by a constant force is the product of the force, the displacement, and the cosine of the angle between them: $W = F\,d\cos\theta$ . The SI unit is the joule (J): 1 J = 1 newton acting through 1 metre in its own direction. Work is a scalar — it has magnitude and sign but no direction.

Work done by a constant force

W = F\,d\cos\theta

Wwork done (joules, J)
Fmagnitude of the applied force (N)
dmagnitude of the displacement (m)
$\theta$ angle between the force and the displacement

Work counts only the part of the force along the displacement. The vertical component F sin θ is perpendicular to motion and does no work.

Worked example

A force of 20 N pulls a box 5 m along the floor, acting at 60° above the horizontal. How much work does the force do?

Practice this conceptself-check · 4 quick reps

From the bank · past-year question

Example 1Work, Energy and PowerEASY

Work is said to be one Joule when a force of

[Q117 · Sep · 2021]

Work needs MOVEMENT in the force's direction — holding a weight is zero work

If the displacement is zero, the work is zero no matter how large the force. Holding a heavy load still, or pushing a wall that does not move, does no physics work even though you feel tired.

Concept 2 of 3

The sign of work — positive, zero, or negative by the angle

Intuition

The cosine in

W = F d\cos\theta

carries the sign. When the force helps the motion the work is positive; when it is at right angles it contributes nothing; when it opposes the motion it is negative (friction removes energy this way). Three angles cover almost every NDA question on work: 0°, 90°, and 180°.

Definition

The sign of work follows the angle between force and displacement:

θ = 0° (parallel): $\cos 0^\circ = 1$ , work is positive maximum — force fully aids the motion.
θ = 90° (perpendicular): $\cos 90^\circ = 0$ , work is zero — e.g. centripetal force, or carrying a load horizontally.
θ = 180° (anti-parallel): $\cos 180^\circ = -1$ , work is negative — e.g. friction, or pulling against the direction of motion.

Sign cases of W = F d cos θ

\cos 0^\circ = 1,\quad \cos 90^\circ = 0,\quad \cos 180^\circ = -1

$\theta = 0^\circ$ force along motion — positive work
$\theta = 90^\circ$ force perpendicular — zero work
$\theta = 180^\circ$ force opposes motion — negative work

Worked example

A frictional force of 6 N acts on a block that slides 3 m in the direction opposite to the friction. How much work does friction do?

Practice this conceptself-check · 4 quick reps

From the bank · past-year question

Example 2Work, Energy and PowerEASY

The work done by the force acting on an object is zero if the displacement of the object

[Q61 · Sep · 2025]

Perpendicular force does ZERO work — not maximum

Carrying a weight at constant height, or the centripetal force on an orbiting body, does no work because the force is at

90^\circ

to the motion. Students often confuse "large force" with "large work" — but with

\cos 90^\circ = 0

the work is exactly zero.

Negative work means the force OPPOSES motion

When force and displacement are anti-parallel (

\theta = 180^\circ

\cos\theta = -1

and the work is negative — the force is taking energy away (friction, air resistance, a brake). It does not mean "no work".

Drill 2 more on the sign of work — positive, zero, or negative by the angle

Concept 3 of 3

Work done by gravity depends only on the height change

Intuition

Gravity is a conservative force, so the work it does on a body moving from one point to another depends only on the change in height — not on the route taken. Climb a hill by a straight path or a winding one, lift the same load up the same height, and gravity does the same work either way.

Definition

The work done by gravity on a body that moves through a vertical height change $h$ is $W = \pm mgh$ , where the sign is negative when the body rises (gravity opposes) and positive when it falls (gravity aids). Crucially this depends only on the vertical displacement, not on the horizontal path — that path-independence is the defining property of a conservative force. So the statement "work done by gravity depends on the path followed" is false.

Work done by gravity over a height change h

W_\text{gravity} = \pm\,mgh

mmass of the body (kg)
gacceleration due to gravity ( $\approx 9.8$ m/s²)
hvertical height change only (m)

Worked example

A 2 kg book is lifted 1.5 m onto a shelf. How much work does gravity do on it? Take

g = 10

m/s².

Practice this conceptself-check · 4 quick reps

From the bank · past-year question

Example 3Work, Energy and PowerEASY

Which one of the following statements for the work done by gravity on a body is NOT correct ?

[Q56 · Sep · 2025]

Gravity's work does NOT depend on the path

Because gravity is conservative, the work it does between two points is fixed by the height difference alone. A statement that "work done by gravity depends on the path followed" is the wrong one the NDA tests.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

What work means in physics — W = F d cos θ
Work done by a constant force
$W = F\,d\cos\theta$
The sign of work — positive, zero, or negative by the angle
Sign cases of W = F d cos θ
$\cos 0^\circ = 1,\quad \cos 90^\circ = 0,\quad \cos 180^\circ = -1$
Work done by gravity depends only on the height change
Work done by gravity over a height change h
$W_\text{gravity} = \pm\,mgh$

Watch out for (4)

Work needs MOVEMENT in the force's direction — holding a weight is zero work
→ What work means in physics — W = F d cos θ
Perpendicular force does ZERO work — not maximum
→ The sign of work — positive, zero, or negative by the angle
Negative work means the force OPPOSES motion
→ The sign of work — positive, zero, or negative by the angle
Gravity's work does NOT depend on the path
→ Work done by gravity depends only on the height change

Drill every past-year question on this subtopic

5 questions from the bank — paginated, with cart and Word-export support.

Drill the 5 questions