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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.

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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 dcos⁡θW = 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 dcos⁡θW = F\,d\cos\theta
  • Wwork done (joules, J)
  • Fmagnitude of the applied force (N)
  • dmagnitude of the displacement (m)
  • \thetaangle between the force and the displacement
FF cos θF sin θθdisplacement dOnly F cos θ is along d, so W = F d cos θ

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?
  1. Use W=F dcos⁡θW = F\,d\cos\theta.
  2. Substitute F=20F = 20 N, d=5d = 5 m, θ=60∘\theta = 60^\circ, with cos⁡60∘=0.5\cos 60^\circ = 0.5.
  3. W=20×5×0.5=50W = 20 \times 5 \times 0.5 = 50 J.
Answer:W=50W = 50 J.
Practice this conceptself-check · 4 quick reps

Try it yourself

A horizontal force of 12 N drags a crate 8 m across a floor in the same direction as the force. How much work is done?

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

  1. 1.
    A 10 N force moves a body 3 m in the direction of the force. Find the work done.
  2. 2.
    What is the SI unit of work?
  3. 3.
    Is work a scalar or a vector quantity?
  4. 4.
    A 50 N force acts at 60° to a 4 m displacement. Find the work done.

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=Fdcos⁡θW = 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∘=1\cos 0^\circ = 1, work is positive maximum — force fully aids the motion.
  • θ = 90° (perpendicular): cos⁡90∘=0\cos 90^\circ = 0, work is zero — e.g. centripetal force, or carrying a load horizontally.
  • θ = 180° (anti-parallel): cos⁡180∘=−1\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∘=1,cos⁡90∘=0,cos⁡180∘=−1\cos 0^\circ = 1,\quad \cos 90^\circ = 0,\quad \cos 180^\circ = -1
  • \theta = 0^\circforce along motion — positive work
  • \theta = 90^\circforce perpendicular — zero work
  • \theta = 180^\circforce 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?
  1. Friction opposes the motion, so the angle between friction and displacement is 180∘180^\circ.
  2. W=F dcos⁡180∘=6×3×(−1)W = F\,d\cos 180^\circ = 6 \times 3 \times (-1).
  3. W=−18W = -18 J — negative, because friction removes energy from the block.
Answer:W=−18W = -18 J (negative work).
Practice this conceptself-check · 4 quick reps

Try it yourself

A satellite moves in a circular orbit. The gravitational pull on it always points toward the centre while its velocity is tangent to the circle. How much work does gravity do on the satellite over one orbit?

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

  1. 1.
    Force and displacement are anti-parallel. Is the work positive, zero, or negative?
  2. 2.
    A force acts perpendicular to the displacement. How much work is done?
  3. 3.
    A man carries a suitcase horizontally across a room. What work does he do against gravity?
  4. 4.
    Force and displacement point the same way. Sign of work?

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∘90^\circ to the motion. Students often confuse "large force" with "large work" — but with cos⁡90∘=0\cos 90^\circ = 0 the work is exactly zero.

Negative work means the force OPPOSES motion

When force and displacement are anti-parallel (θ=180∘\theta = 180^\circ), cos⁡θ=−1\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 hh is W=±mghW = \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

Wgravity=± mghW_\text{gravity} = \pm\,mgh
  • mmass of the body (kg)
  • gacceleration due to gravity (≈9.8\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=10g = 10 m/s².
  1. The book rises, so gravity opposes the motion — the work it does is negative.
  2. Wgravity=−mgh=−(2)(10)(1.5)W_\text{gravity} = -mgh = -(2)(10)(1.5).
  3. Wgravity=−30W_\text{gravity} = -30 J (and you do +30+30 J of work against gravity).
Answer:Wgravity=−30W_\text{gravity} = -30 J.
Practice this conceptself-check · 4 quick reps

Try it yourself

A 5 kg mass slides down a frictionless ramp, dropping a vertical height of 2 m. How much work does gravity do? Take g=10g = 10 m/s².

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

  1. 1.
    Does the work done by gravity depend on the path taken between two points?
  2. 2.
    A 3 kg body falls 4 m. Work done by gravity? (g = 10)
  3. 3.
    A body is lifted up. Is the work done BY gravity positive or negative?
  4. 4.
    A ball is thrown up and returns to the same point. Net work done by gravity?

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

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Formulas (3)

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Mastery check — 2 interleaved questions

Try each one before clicking. Questions are interleaved across the concepts above, not grouped — interleaving sharpens transfer.

Example 1Work, Energy and PowerMODERATE
A mass M is dragged by a pulley on a horizontal plane by a force anti-parallel to its displacement. The work done in pulling the mass M is

[Q79 · Apr · 2022]

Example 2Work, Energy and PowerEASY
A negative work is done when an applied force F and the corresponding displacement S are

[Q150 · Sep · 2021]

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