Work Done by a Constant Force

Overview

In physics, work is done when a force causes an object to move through a displacement.

Work depends on:

• Magnitude of the force
• Magnitude of displacement
• Angle between force and displacement

For a constant force:

W = F⃗ · d⃗

or:

W = Fd cos θ

Learning Outcomes

Students should be able to:

• Define work scientifically.
• Explain why work is a scalar quantity.
• Calculate work done by a constant force.
• Resolve force into parallel and perpendicular components.
• Differentiate between positive, negative and zero work.
• Calculate work done by several forces.
• State the SI unit and dimensions of work.

1. Definition of Work

Work done on an object is the scalar or dot product of force and displacement:

W = F⃗ · d⃗

Its magnitude is:

W = Fd cos θ

Where:

• W = work done
• F = magnitude of constant force
• d = magnitude of displacement
• θ = angle between force and displacement

Work is a scalar quantity because a dot product produces a scalar.

2. Conditions Necessary for Work

For mechanical work to be done:

• A force must act on the object.
• The object must undergo displacement.
• The force must have a component along the displacement.

If any of these conditions is missing, the work done is zero.

3. Effective Component of Force

Suppose a force F acts at angle θ to horizontal displacement d.

The force has two components:

Parallel component

F_∥ = F cos θ

This component acts along the displacement and performs work.

Perpendicular component

F_⊥ = F sin θ

This component is perpendicular to the displacement and performs no work.

Therefore:

W = F_∥ d W = (F cos θ) d

Work Done by a Constant Force

Intermediate Level

4. Positive Work

Work is positive when force has a component in the direction of displacement.

The angle lies in the range:

0° ≤ θ < 90°

In this range:

cos θ > 0

Therefore:

W > 0

Maximum positive work

When force and displacement act in the same direction:

θ = 0° W = Fd cos 0° W = Fd

Examples include:

• Pulling a box forward
• Gravity acting on a falling object
• An engine moving a vehicle forward

5. Negative Work

Work is negative when force has a component opposite to displacement.

The angle lies in the range:

90° < θ ≤ 180°

In this range:

cos θ < 0

Therefore:

W < 0

When force acts directly opposite to displacement:

θ = 180° W = Fd cos 180° W = −Fd

Examples include:

• Friction acting on a sliding object
• Braking force acting on a moving vehicle
• Air resistance opposing motion

Negative work generally removes mechanical energy from an object.

6. Zero Work

Work is zero when force is perpendicular to displacement:

θ = 90°

Therefore:

W = Fd cos 90° W = 0

Work is also zero when:

F = 0

or:

d = 0

Examples include:

• Normal reaction on an object moving horizontally
• Gravitational force on an object moving horizontally
• Holding a stationary object
• Centripetal force in uniform circular motion

7. Work and Angle

Work and Angle

8. Net Work Done by Several Forces

If several constant forces act on an object, calculate the work performed by each force separately.

W₁ = F₁ d cos θ₁ W₂ = F₂ d cos θ₂ W₃ = F₃ d cos θ₃

The net work is:

W_net = W₁ + W₂ + W₃ + ⋯

or:

W_net = Σᵢ Fᵢ d cos θᵢ

For horizontal displacement:

• Horizontal forces may perform positive or negative work.
• Weight performs zero work.
• Normal reaction performs zero work.

9. SI Unit of Work

From:

W = Fd

The unit of force is newton and the unit of displacement is metre.

Therefore:

[W] = N m

The SI unit of work is the joule:

1 J = 1 N m

Using:

1 N = 1 kg m s⁻²

Therefore:

1 J = 1 kg m s⁻² × 1 m 1 J = 1 kg m² s⁻²

10. Dimensions of Work

W = Fd

Dimensions of force are:

[F] = [MLT⁻²]

Therefore:

[W] = [MLT⁻²][L] [W] = [ML²T⁻²]

Work and energy have the same unit and dimensions because work represents energy transferred by a force.

11. Work Is Not the Same as Effort

In everyday language, holding a heavy object may feel like hard work. However, if the object does not move:

d = 0

Therefore:

W = F(0) cos θ W = 0

Mechanical work may be zero even though the person uses biological energy.

Formula Summary

Formula Summary

Solved Examples

Example 1: Force Along Displacement

A constant force of 20 N moves a box through 5 m in the same direction. Calculate the work done.

Given:

F = 20 N d = 5 m θ = 0°

Formula:

W = Fd cos θ

Solution:

W = (20)(5) cos 0° W = (20)(5)(1) W = 100 J

Final answer:

W = 100 J

Example 2: Force Acting at an Angle

A 50 N force moves an object through 3 m. The force makes an angle of 60° with the displacement.

Formula:

W = Fd cos θ

Solution:

W = (50)(3) cos 60° W = (50)(3)(0.5) W = 75 J

Final answer:

W = 75 J

Example 3: Negative Work by Friction

A frictional force of 8 N opposes an object moving through 6 m.

Given:

F = 8 N d = 6 m θ = 180°

Formula:

W = Fd cos θ

Solution:

W = (8)(6) cos 180° W = (8)(6)(−1) W = −48 J

Final answer:

W = −48 J

Example 4: Textbook Suitcase Example

A suitcase is pulled through 4.90 m by a constant force of 16.8 N acting at 30° above the horizontal. A frictional force of 0.170 N opposes its motion.

Part A: Work done by applied force

W_M = F_M d cos θ_M W_M = (16.8)(4.90) cos 30° W_M = (16.8)(4.90)(0.866) W_M = 71.3 J W_M = 71.3 J

Part B: Work done by friction

W_f = F_f d cos θ_f W_f = (0.170)(4.90) cos 180° W_f = (0.170)(4.90)(−1) W_f = −0.833 J W_f = −0.833 J

Part C: Work by weight and normal reaction

Both forces are perpendicular to horizontal displacement:

θ = 90°

Therefore:

W_g = 0 W_N = 0

Part D: Net work

W_net = W_M + W_f + W_g + W_N W_net = 71.3 − 0.833 + 0 + 0 W_net = 70.467 J W_net ≈ 70.5 J

Scientist Connection

The SI unit of work is named after James Prescott Joule, whose experiments helped establish that heat is a form of energy transfer.

1 joule = 1 newton-metre

Important Exam Facts

• Work is the dot product of force and displacement.
• Work is a scalar quantity.
• Only F cos θ performs work along displacement.
• Work is positive when force supports motion.
• Work is negative when force opposes motion.
• Work is zero when force and displacement are perpendicular.
• No displacement means no mechanical work.
• Net work is the sum of work done by all forces.
• The SI unit is joule.
• The dimensions are [ML²T⁻²].

Common Mistakes

• Using W = Fd when the force acts at an angle.
• Using F sin θ instead of F cos θ.
• Measuring θ from the wrong direction.
• Ignoring the negative sign of frictional work.
• Thinking every acting force performs work.
• Including weight in horizontal motion without checking the angle.
• Confusing work with force or power.
• Writing joule per second as the unit of work.
• Forgetting that work is a scalar.

MCQs with Answers

1. Work done by a constant force is: A. Fd sin θ B. Fd cos θ C. Fd D. Fd² Answer: B — W = Fd cos θ.

2. Work is a: A. Vector quantity B. Scalar quantity C. Base quantity D. Dimensionless quantity Answer: B — Work is obtained from a dot product.

3. Work is zero when force and displacement are: A. Parallel B. Opposite C. Perpendicular D. Equal Answer: C — cos 90° = 0.

4. Friction usually performs: A. Positive work B. Negative work C. Zero work in every case D. Infinite work Answer: B — Friction generally acts opposite to displacement.

5. The SI unit of work is: A. Watt B. Newton C. Joule D. Pascal Answer: C — Joule.

Short-Question Preparation

Short-Question Preparation

Long-Question Preparation

A complete answer should include:

• Definition of work.
• Vector expression W = F⃗ · d⃗.
• Scalar expression W = Fd cos θ.
• Meaning of F, d and θ.
• Resolution of force into components.
• Explanation of F cos θ.
• Explanation of why F sin θ does no work.
• Conditions for positive work.
• Conditions for negative work.
• Conditions for zero work.
• Maximum positive work.
• Maximum negative work.
• SI unit of work.
• Relationship 1 J = 1 N m.
• Dimensional formula.
• Work done by several forces.
• Definition of net work.
• Suitable real-life examples.

Sticky Notes

Sticky Notes

• Main formula: W = Fd cos θ
• Component rule: Only the force component parallel to displacement performs work.
• Sign rule: Same direction is positive; opposite direction is negative.
• Zero-work rule: Perpendicular force or zero displacement means zero work.
• Exam warning: Always identify the angle between force and displacement.

Ask-the-Tutor Modes

Students may select:

• Explain work in simpler words.
• Determine whether work is positive, negative or zero.
• Resolve force into useful components.
• Check a numerical solution one step per line.
• Explain why a particular force performs no work.

60-Second Revision

W = F⃗ · d⃗ W = Fd cos θ 1 J = 1 N m [W] = [ML²T⁻²] θ = 0°: maximum positive work θ = 90°: zero work θ = 180°: maximum negative work Work is scalar. Net work is the sum of work done by all forces.