Work in AP Physics C: Mechanics ⚙️

students, imagine pushing a shopping cart across a store floor. If you push hard but the cart does not move, did you do any work in physics? What if you push with the same force and the cart rolls forward? In AP Physics C: Mechanics, work is the link between a force and a change in motion or energy. This lesson will help you understand what work means, how to calculate it, and why it matters in the larger topic of Work, Energy, and Power.

Lesson Goals

By the end of this lesson, students, you should be able to:

explain the main ideas and vocabulary of work
calculate work from force and displacement
use the correct sign of work in common situations
connect work to energy changes and power
solve AP Physics C: Mechanics problems involving work using evidence and examples ✅

What Work Means in Physics

In everyday speech, “work” can mean effort. In physics, work has a specific meaning. Work is done when a force causes an object to move through a displacement.

For a constant force, the work done by the force is

$$W = Fd\cos\theta$$

where:

$W$ is work
$F$ is the magnitude of the force
$d$ is the magnitude of the displacement
$\theta$ is the angle between the force and displacement vectors

This equation shows an important idea: only the part of the force in the direction of motion contributes to work. If the force is exactly along the displacement, then $\theta = 0^\circ$ and $\cos\theta = 1$, so the work is positive and largest. If the force is perpendicular to the displacement, then $\theta = 90^\circ$ and $\cos\theta = 0$, so the work is zero.

Example: pushing a cart 🛒

Suppose you push a cart with a force of $50\ \text{N}$ for a distance of $4\ \text{m}$ in the same direction as motion. Then

$$W = (50)(4)\cos 0^\circ = 200\ \text{J}$$

So the work done is $200\ \text{J}$.

If the same force is applied at an angle of $60^\circ$ above the direction of motion, then

$$W = (50)(4)\cos 60^\circ = 100\ \text{J}$$

Less work is done in the direction of motion because part of the force is upward, not forward.

Signs, Direction, and Meaning

Work can be positive, negative, or zero. The sign depends on the angle between force and displacement.

Positive work happens when force helps the motion. This usually increases the object’s kinetic energy.
Negative work happens when force opposes the motion. This usually decreases the object’s kinetic energy.
Zero work happens when force is perpendicular to displacement, or when there is no displacement.

Positive work

If you pull a sled forward and it moves forward, your force does positive work.

Negative work

If friction acts opposite the motion, friction does negative work. For example, when a book slides across a table, kinetic friction removes mechanical energy from the book’s motion and turns some of it into thermal energy.

Zero work

The normal force on a box resting on a horizontal floor does zero work if the box moves horizontally, because the normal force is vertical while the displacement is horizontal.

A very important reminder for students: work depends on both force and displacement. A large force can still do zero work if there is no displacement, like pushing on a wall that does not move.

Work as an Energy Transfer

Work is one way energy is transferred between objects or between a force and an object. This is why work belongs in the larger topic of Work, Energy, and Power.

The work-energy theorem says

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

where $W_{\text{net}}$ is the net work done on an object and $\Delta K$ is the change in kinetic energy.

This theorem is powerful because it connects force and motion without requiring you to track time directly. If the net work is positive, kinetic energy increases. If the net work is negative, kinetic energy decreases.

Example: speeding up a cyclist 🚴

A cyclist’s pedal force does positive work on the bicycle, increasing its speed. Air resistance and friction do negative work, reducing the energy gain. The net work determines the final change in kinetic energy.

If the net work on the cyclist and bike system is $300\ \text{J}$, then

$$\Delta K = 300\ \text{J}$$

The kinetic energy increases by $300\ \text{J}$.

Calculating Work with Variable Forces

Sometimes the force is not constant. In AP Physics C: Mechanics, you should be ready to handle this case using calculus.

If force varies with position along the line of motion, the work is

$$W = \int_{x_i}^{x_f} F(x)\,dx$$

This means work is the area under a force-versus-position graph.

Example: stretching a spring 🪀

A spring follows Hooke’s law:

$$F(x) = -kx$$

The negative sign shows the spring force points opposite the displacement from equilibrium. If you slowly stretch a spring from $x = 0$ to $x = x_f$, the work done by an external agent is

$$W_{\text{ext}} = \int_0^{x_f} kx\,dx = \frac{1}{2}kx_f^2$$

This result is very important in mechanics. It shows that the energy stored in a stretched or compressed spring is

$$U_s = \frac{1}{2}kx^2$$

So work and energy are closely connected.

Reading a force graph

If a force increases linearly from $0$ to $10\ \text{N}$ over a displacement of $2\ \text{m}$, the work is the area of a triangle:

$$W = \frac{1}{2}(2)(10) = 10\ \text{J}$$

That method is often faster than using the integral when the graph has a simple shape.

Work Done by Gravity and Friction

Two forces appear often in AP Physics C: Mechanics: gravity and friction.

Gravity

Near Earth’s surface, the gravitational force on an object is

$$F_g = mg$$

If an object moves upward a distance $h$, gravity does negative work:

$$W_g = -mgh$$

If the object moves downward a distance $h$, gravity does positive work:

$$W_g = mgh$$

This is one reason gravitational potential energy is useful. The work done by gravity is related to the change in gravitational potential energy by

$$W_g = -\Delta U_g$$

where

$$U_g = mgh$$

Friction

Kinetic friction usually acts opposite the direction of motion, so its work is negative:

$$W_f = -f_k d$$

because the friction force and displacement point in opposite directions.

Example: if $f_k = 8\ \text{N}$ and the object slides $5\ \text{m}$,

$$W_f = -(8)(5) = -40\ \text{J}$$

That means $40\ \text{J}$ of mechanical energy is removed from the object’s motion, usually becoming thermal energy.

Common AP Physics C Problem-Solving Tips

students, many work problems become easier if you follow a clear process:

Identify the force or forces doing work.
Find the displacement.
Determine the angle $\theta$ between force and displacement.
Use $W = Fd\cos\theta$ for constant forces.
Use $W = \int F(x)\,dx$ if the force changes with position.
Use $W_{\text{net}} = \Delta K$ when the problem asks about speed or kinetic energy.

Example: box pulled across a floor

A $10\ \text{kg}$ box is pulled $3\ \text{m}$ by a horizontal force of $20\ \text{N}$ on a rough floor. Friction is $5\ \text{N}$ opposite the motion.

Work by the pulling force:

$$W_p = (20)(3) = 60\ \text{J}$$

Work by friction:

$$W_f = -(5)(3) = -15\ \text{J}$$

Net work:

$$W_{\text{net}} = 60 - 15 = 45\ \text{J}$$

So the box’s kinetic energy increases by $45\ \text{J}$.

How Work Connects to Power

Work tells you how much energy is transferred. Power tells you how fast that transfer happens.

$$P = \frac{W}{t}$$

The SI unit of power is the watt, where

$$1\ \text{W} = 1\ \text{J/s}$$

Two people can do the same amount of work, but the one who does it in less time has greater power. For example, if two students lift identical backpacks to the same height, the one who does it faster has greater power output.

Conclusion

Work is one of the core ideas in AP Physics C: Mechanics because it connects force, motion, and energy. When a force acts along a displacement, it can transfer energy and change an object’s kinetic energy. Positive work adds energy, negative work removes energy, and zero work means no energy transfer from that force. Whether you are analyzing a cart, a spring, gravity, or friction, the same principles apply. Mastering work will help you understand energy conservation, potential energy, and power throughout the rest of the unit 🚀

Study Notes

Work is done when a force causes displacement.
For a constant force, $W = Fd\cos\theta$.
Only the component of force parallel to displacement does work.
Positive work increases kinetic energy; negative work decreases kinetic energy.
Zero work happens when force is perpendicular to displacement or when displacement is zero.
The work-energy theorem is $W_{\text{net}} = \Delta K$.
For variable force, use $W = \int_{x_i}^{x_f} F(x)\,dx$.
The work done by gravity near Earth is $W_g = -mgh$ for upward motion.
Kinetic friction usually does negative work: $W_f = -f_k d$.
Power is work per time: $P = \frac{W}{t}$.
Work is a major bridge between forces and energy in AP Physics C: Mechanics.