AP Physics C: Mechanics · Unit 3: Work, Energy, and Power · Lesson 3.2

Deep Dive: Work

🔬 Deep Dive

This is your textbook for this topic. Take your time. Read it more than once.

3.2.A.13.2.A.1.i–vConcept

What Is Work? Conservative vs. Nonconservative Forces

Work is the amount of energy transferred into or out of a system by a force acting over a distance. But not all forces transfer energy the same way — the distinction between conservative and nonconservativeforces is one of the most important ideas in this entire unit.

🔑Work done by a conservative force — gravity, spring force — depends only on where you started and where you ended up, never on the path taken in between. If the system returns to its starting configuration, a conservative force's total work (and the corresponding change in potential energy) is exactly zero. This path-independence is exactly what makes potential energypossible to define at all — it only exists for conservative forces.

⚠️Work done by a nonconservative force — friction, air resistance (the two most common examples) — genuinely depends on the path. A longer, more winding path means more energy lost to friction, even between the same two endpoints. There's no potential energy associated with these forces.

3.2.A.2Concept

Work Is a Scalar

Like kinetic energy, work is a scalar — but unlike kinetic energy, work can be positive, negative, or zero. Positive work adds energy to a system; negative work removes it; zero work means the force present didn't transfer any energy at all, even if it was doing something.

💡A common example of zero work: a satellite in a perfectly circular orbit. Gravity is very much acting on it, but since gravity is always perpendicular to the satellite's velocity, it does zero work — the satellite's speed (and kinetic energy) never changes.

3.2.A.33.2.A.3.i–ivMath

The Work Integral and the Dot Product

When a force varies as an object moves, work is calculated with an integral over the path from point a to point b:

W = ∫(a to b) F⃗(r⃗) · dr⃗

The dot (·) here is the dot product — a way of combining two vectors into a single scalar:

A⃗ · B⃗ = AB cos θ

🔑This single fact controls everything about work: only the component of force parallel to displacement changes a system's energy. A force component perpendicular to displacement can redirect motion — change its direction — without doing any work or changing kinetic energy at all.

If the parallel component of the force happens to stay constant over the whole displacement, the integral collapses into simple multiplication:

W = Fd cos θ

Explore the angle-dependence directly below.

A constant 20 N force acts on a block as it moves 8 m to the right. Adjust the angle between the force and the displacement — watch work swing from strongly positive, to zero, to strongly negative.

angle θ30°

W = Fd cos θ = (20)(8)cos(30°) = 138.6 J

The force has a component along the displacement — positive work, adding energy to the system.

When the force genuinely changes with position, you need the full integral — which is exactly the area under a force-vs-position graph. Explore that below.

A force that varies with position, F(x) = 4√x. Set the interval boundaries — the shaded area equals the work done by this force over that interval.

x₁1.0 m

x₂6.0 m

W = ∫[1.0 to 6.0] F(x) dx ≈ 36.53 J

This is exactly the same idea as area under a velocity-time graph giving displacement (Lesson 1.3) — area under a force-position graph gives work.

ExampleGuided Example — Work at an Angle

A 15 N force is applied to a crate at 40° above the horizontal while the crate slides 6 m horizontally across the floor. Find the work done by this force.

Step 1 — Identify what's parallel to the displacement

The crate moves horizontally, so only the horizontal component of the applied force does work.

3.2.A.43.2.A.4.i–iiiMath

The Work-Energy Theorem

The work-energy theorem ties everything in this lesson back to Lesson 3.1: the change in an object's kinetic energy equals the total (net) work done by every force acting on it.

ΔK = W(net) = Σ Fᵢd∥,ᵢ

🔑This gives you a second route to finding an object's speed — one that doesn't require kinematics at all. If you know the net work done on an object, you know exactly how much its kinetic energy changed, full stop.

A useful special case: if a system's center of mass and the point where a force is actually applied both move the same distance, you can treat the whole system as a single object, and only its kinetic energy changes as a result.

Friction is a particularly important nonconservative case. The mechanical energy dissipated by friction is:

ΔE(mech) = Ff · d cos θ

ExampleWorked Example — Work-Energy Theorem with Friction

A 3 kg block starts at rest and is pushed 4 m across a rough floor by a constant 20 N horizontal force. Friction exerts 8 N opposing the motion. Find the block's final speed using the work-energy theorem.

3.2.A.5Math

Work as Area Under a Curve

Since work is defined by an integral, it's also — exactly as with displacement from a velocity-time graph back in Lesson 1.3 — the area under a graph: specifically, the area under a graph of F∥ (the force component parallel to displacement) plotted against displacement.

💡This is often the fastest way to find work on the AP exam: given a force-vs-position graph, you don't need to know the underlying function at all — just find the area geometrically (triangles, rectangles, trapezoids) between the curve and the axis.

← Back to Lesson 3.2Ready for 3.3? Potential Energy formalizes the "stored" energy side of every conservative force from this lesson.