Work, Energy, and Power

What this unit covers

The topics below follow the published Physics C: Mech course framework for Unit 3. This unit is worth 15–25% of the exam, so budget your time against that rather than against how long the unit takes to teach.

Lessons in this unit

• Work & the Work-Energy Theorem14 min · 3 objectivesDefine work as the line integral W = ∫F·dx and compute it for a variable force · State and apply the work-energy theorem W_net = ΔKE · Compute the work done by a position-dependent force such as a spring
• Potential Energy & the Force-Energy Relationship14 min · 3 objectivesDefine potential energy for a conservative force as U = −∫F dx · Recover the force from a potential energy function using F = −dU/dx · Read a potential energy curve to locate equilibria and judge their stability
• Conservation of Energy14 min · 3 objectivesState conservation of mechanical energy for systems acted on only by conservative forces · Solve motion problems using KE_i + U_i = KE_f + U_f · Account for energy removed by friction using the work of nonconservative forces
• Power12 min · 3 objectivesDefine power as the rate of doing work, P = dW/dt · Compute instantaneous power as the dot product P = F·v · Distinguish average power from instantaneous power
• Work by a Variable Force: The Integral Definition14 min · 3 objectivesCompute work as the integral of force over displacement · Interpret work as the area under a force–position graph · Derive elastic potential energy from the work done by a spring
• Potential Energy Curves & the Stability of Equilibrium14 min · 3 objectivesRecover force from a potential energy function using F = −dU/dx · Identify equilibrium points and classify them as stable, unstable or neutral · Use a potential energy diagram with total energy to find turning points
• Gravitational Potential Energy Beyond mgh & Escape Speed14 min · 3 objectivesDerive U = −GMm/r by integrating the gravitational force · Explain the sign convention and why U is negative for bound systems · Compute escape speed and relate total energy to whether an orbit is bound
• Nonconservative Forces & Energy Accounting14 min · 3 objectivesDistinguish conservative from nonconservative forces by the path-independence test · Apply the generalized work–energy theorem including dissipated energy · Track energy through a multi-stage problem without losing a term

Formulas in Unit 3

Every term in Unit 3

All 53 terms we publish for Work, Energy, and Power, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.

Instantaneous power

P = dW/dt = F·v. Average power is total work divided by total time, and the two differ whenever force or speed varies.

Work as an integral

W = ∫F·dr. For a varying force, this is the area under the force-position curve, and F·d is only the constant-force case.

Work-energy theorem

W_net = ΔKE. Derived by integrating ΣF = ma with respect to position.

Force from potential energy

F = −dU/dx. Force points down the potential energy gradient, which is why objects move toward potential minima.

Potential energy curves

Minima are stable equilibria and maxima unstable. The slope gives the force, and the difference between total energy and U gives the kinetic energy.

Turning points

Where total energy equals potential energy, so kinetic energy is zero and the object reverses direction.

Gravitational potential energy in general

U = −GMm/r, taking zero at infinity. The near-Earth mgh is the small-Δr approximation of this.

Escape velocity

v_esc = √(2GM/R), found by setting total mechanical energy to zero. Independent of the escaping object's mass.

Spring potential energy

U = ½kx², the integral of the restoring force kx over displacement.

Conservative forces and path independence

A force is conservative when its work around any closed loop is zero, which is exactly when a potential energy function can be defined.

When to use energy rather than forces

Energy avoids solving for time and handles varying forces and curved paths; forces are needed when time or an individual force is asked for.

Work done by friction

Negative and equal to −f·d where d is the PATH LENGTH, not the displacement. Friction is not conservative, so the path matters.

Work done by a variable force from a graph

The signed area under the force-position curve. Area below the axis represents negative work.

Stability from a potential curve

A minimum of U is stable equilibrium, a maximum unstable, and a flat region neutral. Small displacements from a minimum produce a restoring force.

Reading kinetic energy from a potential curve

KE = E − U at each position. Where the horizontal total-energy line meets the curve, KE is zero and the object turns around.

Deriving force from U(x)

F = −dU/dx, so the force is the negative slope of the potential energy curve. A steeper curve means a stronger force.

Escape energy

The minimum energy to reach infinity with zero speed, so ½mv² = GMm/R. Independent of launch direction for a non-rotating body.

Power delivered by a constant force

P = Fv, so a vehicle at constant speed against constant drag draws constant power while doing zero net work.

Energy in a spring-mass system

Total energy ½kA² is constant. At displacement x, KE = ½k(A² − x²).

Work as a dot product

W = F · d = Fd cos θ for a constant force. Only the component of force along the displacement contributes, so a perpendicular force does no work however large it is.

The sign of work

Positive when the force has a component along the motion, negative when it opposes it. Friction on a sliding block does negative work; the force lifting a box does positive work while gravity does negative.

Work by gravity is path independent

W = −mgΔh regardless of the route taken, which is precisely what makes gravity conservative and allows a potential energy function to exist.

Zero work from perpendicular forces

The normal force on a level surface, the tension in a conical pendulum, and gravity on a circular orbit all do exactly zero work, because F and displacement are perpendicular at every instant.

Net work and the work–energy theorem

W_net = ΔK. The theorem uses the NET work of all forces, so a box lifted at constant speed has zero net work even though gravity and the lifter each do a great deal individually.

Kinetic energy is a scalar

K = ½mv² has no direction and is never negative. Two objects moving oppositely have kinetic energies that add rather than cancel — unlike their momenta.

Kinetic energy in terms of momentum

K = p²/2m. Useful whenever momentum is conserved and energy is asked for, and it shows that at equal momentum the lighter object carries more kinetic energy.

Elastic potential energy

U = ½kx², measured from the spring's natural length. The one-half arises from integrating a force that grows linearly from zero.

Gravitational potential energy near the surface

U = mgh, valid only while g is effectively constant. Over astronomical distances it must be replaced by U = −GMm/r.

Choosing the zero of potential energy

The reference point is arbitrary because only differences in U are physical. Choose it to simplify the algebra, state the choice, and keep it fixed throughout the problem.

Mechanical energy

E = K + U. It is conserved when only conservative forces do work, which is the condition to check rather than assume.

When mechanical energy is conserved

Only if no nonconservative force does work. Friction, drag and applied pushes all break it; normal forces and tensions perpendicular to motion do not, since they do no work.

Nonconservative work

W_nc = ΔE_mech. The work done by nonconservative forces equals the change in mechanical energy, which is the general statement that reduces to conservation when W_nc = 0.

Thermal energy from friction

ΔE_th = f_k d where d is the total PATH LENGTH, not the displacement. An object that slides out and back dissipates over the whole trip even though its displacement is zero.

Energy bar charts

Draw K, U_g, U_s and E_th as bars at the initial and final instants. The total must match, and the visual makes a missing term obvious in a way an equation often does not.

Power as the rate of doing work

P = dW/dt, measured in watts. A machine that does the same work in half the time delivers twice the power, which is why power rather than work is the specification on an engine.

Instantaneous power as force dotted with velocity

P = F · v. At constant speed against drag, the required power grows as the cube of the speed for quadratic drag, since F ∝ v².

Average power

P_avg = W/Δt = ΔE/Δt. It differs from instantaneous power whenever force or speed varies, and questions frequently rely on the distinction.

Efficiency

Useful output energy divided by total input, always less than one for a real machine. The remainder appears as thermal energy, which is where the second law enters mechanics.

Units of energy and power

The joule is a newton-meter and a watt is a joule per second. Checking that an expression reduces to kg·m²/s² catches most algebraic slips.

Spring plus gravity

For a vertical spring, U_total = ½kx² + mgy. Measuring from the hanging equilibrium absorbs the gravitational term into a shifted quadratic, which is why the oscillation frequency is unchanged by gravity.

Equilibrium from the potential curve

Equilibrium occurs where dU/dx = 0, since F = −dU/dx. Zero slope means zero force, but says nothing yet about stability.

Second derivative test for stability

d²U/dx² > 0 at a minimum means stable; < 0 at a maximum means unstable; zero over an interval means neutral. Stability is about curvature, not about the value of U.

Turning points from a U diagram

Where the horizontal total-energy line meets the U curve, K = 0 and the motion reverses. The particle is confined to regions where U ≤ E.

Bound versus unbound motion

A particle trapped between two turning points is bound and oscillates; one whose energy exceeds every barrier escapes. For gravity the dividing line is E = 0.

Deriving escape speed

Set the total energy to zero at the surface: ½mv² − GMm/R = 0 gives v = √(2GM/R), about 11.2 km/s for Earth. Zero total energy is the marginal case for reaching infinity.

Escape speed is mass independent

The escaping object's mass cancels, so a pebble and a spacecraft require the same speed. Only the central body's mass and radius matter.

Total energy of an elliptical orbit

E = −GMm/2a, with a the semi-major axis. The energy depends only on a, so all orbits with the same semi-major axis have the same energy and the same period regardless of eccentricity.

Circular orbit energy relations

K = +GMm/2r, U = −GMm/r and E = −GMm/2r, so E = −K and E = U/2. Speeding a satellite up in place therefore raises it to a larger, SLOWER orbit.

Binding energy

The energy that must be supplied to move a bound object to infinity, equal to |E|. For a circular orbit it is GMm/2r — half the escape energy from that radius.

Work to change orbits

The work required equals the difference in total energy, GMm/2(1/r₁ − 1/r₂). Because E is negative, moving to a higher orbit increases E toward zero and requires positive work.

Why orbital speed falls with radius

v = √(GM/r), so a higher orbit is slower. The satellite gains potential energy faster than it loses kinetic, which is why total energy still rises.

Finding ω by differentiating the total energy

Write E = ½m(dx/dt)² + ½kx², set dE/dt = 0 and the SHM equation drops out. This route is often faster than force analysis for rolling or rotating oscillators.

Where the factor of one half comes from

½kx², ½mv² and ½Iω² all arise from integrating a quantity that grows linearly from zero — force with displacement, momentum with velocity, angular momentum with angular velocity.

What examiners penalize here

• Move fluently in both directions: integrate a conservative force to get U (U = −∫F dx), and differentiate U to get the force (F = −dU/dx). On a U-versus-x graph, remember the force points downhill and equilibria sit where the slope is zero.
• Use P = W/Δt when you know a total amount of work over an interval, and P = Fv (or F·v) when you want the power at a specific instant or speed. A force perpendicular to the velocity — like the centripetal force on a satellite — does zero work and delivers zero power.
• A force perpendicular to displacement does zero work. That single fact disposes of the normal force on a level surface, the tension in a conical pendulum, the magnetic force on a charge, and the centripetal force in any circular orbit — all common exam targets.
• Free-response energy questions are graded on the **accounting**, not the arithmetic. Write an explicit initial-equals-final statement naming every term before substituting numbers. A correct number arrived at without a stated energy equation routinely loses the setup points; a stated equation with a small numerical slip usually keeps most of them.

Practice Physics C: Mech

Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.

Questions about this unit

All 7 units of AP Physics C: Mechanics

1. Unit 1 · Kinematics
2. Unit 2 · Force and Translational Dynamics
3. Unit 3 · Work, Energy, and Power
4. Unit 4 · Linear Momentum
5. Unit 5 · Torque and Rotational Dynamics
6. Unit 6 · Energy and Momentum of Rotating Systems
7. Unit 7 · Oscillations