1. Definition

Work is the transfer of energy that occurs when a force acts on an object while the object undergoes a displacement.

In mechanics, work measures how effectively a force changes an object’s energy.

2. Work Done by a Constant Force

For a constant force acting through a displacement:

\(W=\vec{F}\cdot\delta\vec{r}\)

Using the definition of the dot product:

\(W=F\delta r\cos\theta\)

where:

• $\backslash (W\backslash )$ = work
• \(F\) = magnitude of the force
• \(\delta r\) = magnitude of the displacement
• \(\theta\) = angle between the force and displacement

3. Physical Meaning of the Angle

The angle determines how much of the force contributes to the motion.

Force Parallel to Motion

\(\theta=0\circ\)

\(W=F\delta r\)

Maximum positive work.

Force Perpendicular to Motion

\(\theta=90\circ\)

\(W=0\)

No work is done.

Force Opposite Motion

\(\theta=180\circ\)

\(W=-F\delta r\)

Negative work.

4. Positive and Negative Work

Positive Work

Occurs when force and displacement point generally in the same direction.

Examples:

• pushing a box forward
• gravity acting on a falling object

Positive work increases kinetic energy.

Negative Work

Occurs when force opposes displacement.

Examples:

• friction
• air resistance

Negative work decreases kinetic energy.

5. Units of Work

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

\(1J=1N\cdot m\)

One joule represents one newton acting through one meter of displacement.

6. Work and Components of Force

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

If:

\(F_x=F\cos\theta\)

then:

\(W=F_x\delta r\)

The perpendicular component contributes no work.

7. Work by Variable Forces

In AP Physics C, forces are often not constant.

For a variable force:

\(W=\int\vec{F}\cdot d\vec{r}\)

This is the calculus definition of work.

8. Work from a Force–Position Graph

For one-dimensional motion:

\(W=\intF(x)dx\)

The work equals the area under the force-versus-position graph.

Positive area represents positive work.

Negative area represents negative work.

9. Examples

Lifting an Object

A person lifts a mass vertically upward.

Applied force and displacement are in the same direction.

\(W=Fd\)

Positive work is done by the person.

Friction on a Sliding Box

Friction acts opposite the displacement.

\(W_f=-fd\)

Friction removes mechanical energy from the system.

Carrying a Box Horizontally

The upward force exerted by your arms is perpendicular to the horizontal displacement.

\(W=0\)

No work is done by the upward support force.

10. Connection to Energy

Work is important because it changes energy.

The Work–Energy Theorem states:

\(W_{net}=\delta K\)

where:

• \(W_{net}\)​ = net work done on the object
• \(\delta K\) = change in kinetic energy

This theorem forms the foundation of energy methods in mechanics.

Summary

Work measures energy transfer caused by a force acting through a displacement.

For a constant force:

\(W=Fd\cos\theta\)

The work expression is based on the dot product:

\(\vec{F}\cdot\delta\vec{r}\)

Key ideas:

• positive work increases energy
• negative work decreases energy
• forces perpendicular to motion do no work
• variable-force work is found using integration