What is Work Done?

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

Work is therefore a way of describing an energy transfer.

For mechanical work to occur:

• A force must act on an object.
• The object must move.
• At least part of the force must act in the direction of the displacement.

For example:

• Lifting a box transfers energy into its gravitational potential energy store.
• Pushing a trolley causes energy to be transferred into its kinetic energy store.
• Friction does work on a sliding object, transferring energy from its kinetic store into internal energy stores.

The SI unit of work is the joule (J).

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Work Done and Energy Transferred

Work done and energy transferred have the same numerical value.

If 50 J of work is done on an object, then 50 J of energy has been transferred.

$\boxed{\text{Work done}=\text{Energy transferred}}$

Work done=Energy transferred​

For example, if a motor does 500 J of work lifting a load, the load gains 500 J of gravitational potential energy, assuming no energy is dissipated.

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Work and Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed.

When work is done:

• Energy leaves one store.
• It is transferred through mechanical or electrical work.
• It enters one or more other stores.

Example: lifting a book

Chemical energy store in muscles

$\rightarrow$

→ mechanical work

$\rightarrow$

→ gravitational potential energy store of the book

The energy does not appear from nowhere. It is transferred from the person's chemical store.

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When is Mechanical Work Done?

Mechanical work is done when a force has a component in the direction of displacement.

Examples include:

• Pushing a box across a floor.
• Pulling a sled with a rope.
• Lifting a mass vertically.
• Compressing a spring.
• Braking a moving bicycle.

In each case, a force acts through a displacement.

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When is No Mechanical Work Done?

No mechanical work is done by a force if:

• There is no displacement.
• The force is perpendicular to the displacement.
• The force is zero.

Example 1: Pushing a Wall

A person pushes hard against a wall, but the wall does not move.

Displacement = 0

Therefore:

$W=0$

W=0

The person's muscles still transfer energy internally, but no mechanical work is done on the wall.

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Example 2: Carrying a Bag Horizontally

A person carries a bag at constant height.

The supporting force is upward, but the displacement is horizontal.

The angle between the force and displacement is

$90^\circ$

90∘.

Since:

$\cos 90^\circ=0$

cos90∘=0

the upward force does no mechanical work on the bag.

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Work Done When Force and Displacement Are Parallel

If a constant force acts in the same direction as the displacement:

$\boxed{W=Fd}$

W=Fd​

where:

• $W$W = work done in joules (J)
• $F$F = force in newtons (N)
• $d$d = displacement in metres (m)

One joule is the work done when a force of one newton moves an object one metre in the direction of the force.

$1\text{ J}=1\text{ N}\cdot\text{m}$

1 J=1 N⋅m

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Worked Example 1

A person pushes a box with a force of 40 N through a distance of 5.0 m.

$W=Fd$

W=Fd

$W=(40)(5.0)$

W=(40)(5.0)

$W=200\text{ J}$

W=200 J

Answer: 200 J of work is done on the box.

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Work Done at an Angle

Sometimes the force and displacement are not in the same direction.

For example, a person may pull a suitcase using a handle angled upward.

Only the component of the force parallel to the displacement does work.

$\boxed{W=Fd\cos\theta}$

W=Fdcosθ​

where

$\theta$

θ is the angle between the force and displacement.

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Worked Example 2

A person pulls a suitcase with a force of 60 N at an angle of

$30^\circ$

30∘ above the horizontal. The suitcase moves 8.0 m horizontally.

$W=Fd\cos\theta$

W=Fdcosθ

$W=(60)(8.0)\cos30^\circ$

W=(60)(8.0)cos30∘

$W\approx416\text{ J}$

W≈416 J

Only the horizontal component of the force contributes to the work done on the suitcase.

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Positive, Negative, and Zero Work

Positive Work

Work is positive when the force acts partly in the same direction as the displacement.

Examples:

• A person pushes a trolley forward.
• Gravity does positive work on a falling object.
• A motor lifts an elevator.

Positive work usually transfers energy into an object's kinetic or potential store.

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Negative Work

Work is negative when the force acts opposite to the displacement.

Examples:

• Friction acts on a sliding box.
• Brakes slow a car.
• Gravity acts on an object being lifted.

Negative work removes energy from one store and transfers it elsewhere.

For example, friction does negative work on a moving object, reducing its kinetic energy while increasing internal energy stores.

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Zero Work

Work is zero when:

• There is no displacement, or
• The force is perpendicular to the displacement.

Examples:

• Holding a heavy object still.
• The normal force on a box moving horizontally.
• The centripetal force in uniform circular motion.

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Work Done by Gravity

When an object falls, gravity does positive work.

Its gravitational potential energy store decreases while its kinetic energy store increases.

When an object is lifted, gravity does negative work because gravity acts downward while displacement is upward.

The lifting force does positive work.

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Work Done Against Friction

When an object moves across a rough surface, friction opposes the motion.

The work done against friction transfers energy into the internal energy stores of:

• The object.
• The surface.
• The surroundings.

This often appears as heating.

Example:

A 25 N friction force acts over a distance of 4.0 m.

$W=Fd$

W=Fd

$W=(25)(4.0)$

W=(25)(4.0)

$W=100\text{ J}$

W=100 J

Therefore, 100 J is transferred from the object's kinetic store into internal energy stores.

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Mechanical Systems

In mechanical systems, work can transfer energy between stores.

Examples include:

Lifting a Load

Chemical or electrical store

$\rightarrow$

→ mechanical work

$\rightarrow$

→ gravitational potential store

Accelerating a Car

Chemical store in fuel

$\rightarrow$

→ mechanical work

$\rightarrow$

→ kinetic store

Compressing a Spring

Chemical store in muscles

$\rightarrow$

→ mechanical work

$\rightarrow$

→ elastic store

Braking

Kinetic store

$\rightarrow$

→ work done by friction

$\rightarrow$

→ internal stores

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Electrical Work

Electrical work occurs when a potential difference moves charge through a circuit.

The electrical work done can be written as:

$W=QV$

W=QV

where:

• $W$W = electrical work or energy transferred (J)
• $Q$Q = charge moved (C)
• $V$V = potential difference (V)

Electrical work transfers energy between stores.

Example: electric motor

Electrical transfer

$\rightarrow$

→ mechanical work

$\rightarrow$

→ kinetic energy store

Example: electric heater

Electrical transfer

$\rightarrow$

→ internal energy store

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Work Done and Kinetic Energy

The work-energy principle states that the net work done on an object equals its change in kinetic energy.

$W_{\text{net}}=\Delta E_k$

Wnet​=ΔEk​

If positive net work is done:

• Kinetic energy increases.
• The object speeds up.

If negative net work is done:

• Kinetic energy decreases.
• The object slows down.

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A Problem-Solving Method

Use these steps when solving work problems:

1. Identify the force.
2. Identify the displacement.
3. Determine the angle between them.
4. Choose$W=Fd$W=Fd or$W=Fd\cos\theta$W=Fdcosθ.
5. Substitute values using SI units.
6. Decide whether the work is positive, negative, or zero.
7. Identify the energy stores involved.

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Worked Example 3: Lifting

A student lifts a 12 kg box vertically through 1.5 m at constant speed.

The lifting force equals the weight:

$F=mg=(12)(9.8)=117.6\text{ N}$

F=mg=(12)(9.8)=117.6 N

Work done:

$W=Fd$

W=Fd

$W=(117.6)(1.5)$

W=(117.6)(1.5)

$W=176.4\text{ J}$

W=176.4 J

The student's chemical energy store decreases, while the box's gravitational potential energy store increases by approximately 176 J.

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Worked Example 4: Force Perpendicular to Motion

A satellite moves in a circular orbit.

Gravity acts toward the centre of the orbit, while the satellite's instantaneous displacement is tangential.

The force and displacement are perpendicular.

$W=Fd\cos90^\circ$

W=Fdcos90∘

$W=0$

W=0

Gravity changes the direction of the satellite's velocity but does not change its speed in a perfectly circular orbit.

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Common Mistakes

Avoid these misconceptions:

• A force alone does not guarantee work is done. There must be displacement.
• Distance and displacement are not always interchangeable. Work depends on displacement in the direction of the force.
• Only the force component parallel to the displacement does work.
• Holding an object still does no mechanical work on the object, even though the person uses energy internally.
• Work is not a form of stored energy. It is a transfer pathway.
• Negative work does not mean energy is destroyed. It means energy is transferred out of the chosen store.

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Key Vocabulary

• Work Done – Energy transferred when a force acts through a displacement.
• Joule (J) – The SI unit of work and energy.
• Displacement – The change in position of an object, including direction.
• Mechanical Work – Energy transferred by a force acting through a displacement.
• Electrical Work – Energy transferred when charge moves through a potential difference.
• Positive Work – Work done by a force acting partly in the direction of displacement.
• Negative Work – Work done by a force acting opposite to displacement.
• Net Work – The total work done by all forces acting on an object.
• Conservation of Energy – The principle that energy cannot be created or destroyed.

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Summary

• Work is done when a force causes displacement and has a component in the direction of that displacement.
• Work done is equal to the amount of energy transferred.
• For a force parallel to displacement,$W=Fd$W=Fd.
• For a force acting at an angle,$W=Fd\cos\theta$W=Fdcosθ.
• No mechanical work is done if there is no displacement or if the force is perpendicular to the displacement.
• Positive, negative, and zero work describe how forces transfer energy between stores.
• Work is a transfer pathway, and all work-energy processes obey the principle of conservation of energy.

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Suggested Images

1. Force, Displacement, and Work ⭐

A three-panel diagram showing:

• Force parallel to displacement: positive work.
• Force opposite displacement: negative work.
• Force perpendicular to displacement: zero work.

Include arrows and the angle

$\theta$

θ between force and displacement.

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2. Pulling a Suitcase at an Angle

A diagram of a suitcase pulled by an angled handle showing:

• Applied force$F$F.
• Horizontal displacement$d$d.
• Angle$\theta$θ.
• Horizontal component$F\cos\theta$Fcosθ.

Include the equation

$W=Fd\cos\theta$

W=Fdcosθ.

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3. Energy Transfers During Lifting

A flow diagram showing:

Chemical store in muscles → mechanical work → gravitational potential store of the object

Include an illustration of a person lifting a box.

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4. Positive, Negative, and Zero Work

A comparison infographic with examples:

• Positive: pushing a trolley forward.
• Negative: friction slowing a box.
• Zero: carrying a bag horizontally at constant height.

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5. Mechanical and Electrical Work

A split image showing:

• A motor lifting a mass: electrical transfer → mechanical work → gravitational store.
• A heater: electrical transfer → internal energy store.

This reinforces that work is an energy transfer process in both mechanical and electrical systems.

__________

The work done is the area under the graph:

W = 0.5(8)(30) + 4(30) = 120 + 120 = 240J