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Physics for ME · Chapter 6 of 16 · Intermediate

Work, Energy, and Power

Energy bookkeeping answers questions that force-by-force analysis makes painful: how far, how fast, how much fuel, how big a motor.

01

Readiness check

From Chapters 3 to 5 and Math Chapter 5. Tick only what you can do closed-notes.

  • Compute net force from an FBD.
  • Use v² = v₀² + 2as confidently.
  • Evaluate ∫F dx for a varying force (Math Chapter 5).
  • Keep joules, watts, and newton-metres distinct.
  • Project a force onto a displacement direction (dot product).
0 or 1 weak itemsContinue with this chapter.
2 weak itemsReview Math Chapter 5; work is its physical twin.
3 or more weak itemsStep back to Chapter 4 and Math Chapter 5.
02

The core idea

Work moves energy between accounts. The totals always balance.

W = F·d (along motion)KE = ½mv²P = W/t = F·v

The work-energy theorem says net work equals the change in kinetic energy. With potential energy (mgh, ½ks²) on the books, conservation turns many force problems into one-line audits: energy in, energy out, energy stored.

The skill works when: you can name every account (kinetic, gravitational, spring, heat via friction) and track the transfers.
The skill breaks down when: a path-dependent loss (friction) is treated as recoverable, or force perpendicular to motion is credited with work.
The concept. Four accounts cover mechanics: motion, height, springs, and the one-way drain to heat. Engineering is managing the transfers.
03

What this chapter covers

  • 6.1 Work by a constant force: W = Fd cos θ, only the along-motion part counts.
  • 6.2 Work by a varying force: the integral ∫F dx (springs).
  • 6.3 Kinetic energy and the work-energy theorem: net work = ΔKE.
  • 6.4 Potential energy: gravity mgh and spring ½ks².
  • 6.5 Conservation of energy: the audit principle.
  • 6.6 Friction as the leak: mechanical energy into heat, one way.
  • 6.7 Power: the rate of doing work; motor and engine sizing.
  • 6.8 Efficiency: useful out over total in.

Engineering connection: Dynamics, Thermodynamics, Machine sizing, Energy Systems.

04

Worked example: braking distance by energy audit

A 1200 kg car at 25 m/s (90 km/h) brakes with a steady total force of 7200 N. Find the braking distance, then state what happens at double the speed.

Figure 1. The governing model: kinetic energy drained by brake-force work over the stopping distance.
  1. ProblemFind the stopping distance in Figure 1.
  2. Given / findm = 1200 kg, v = 25 m/s, F = 7200 N steady. Find d; then d at 50 m/s.
  3. AssumptionsLevel road, constant braking force, all kinetic energy into brake heat.
  4. ModelWork-energy theorem: the brakes must do work equal to the kinetic energy.
  5. Equations½mv² = F·d
  6. SolveKE = 0.5 × 1200 × 625 = 375 000 J. d = 375 000/7200 = 52.1 m. At 50 m/s: KE quadruples (1.5 MJ), so d = 208 m.
  7. CheckForce route: a = 7200/1200 = 6 m/s²; d = v²/2a = 625/12 = 52.1 m. Same answer, two methods. The v² scaling matches Chapter 1's lesson.
  8. ConclusionDouble the speed, four times the distance: the single most consequential equation in road safety, and the same audit sizes brakes, crash barriers, and flywheel absorbers.
Result. d = 52.1 m at 25 m/s; 208 m at 50 m/s. Energy and force methods agree.
05

Misconceptions and diagnostics

MistakeSymptomDiagnostic questionCorrection
Perpendicular forces credited with workNormal force or centripetal force "doing work""Does the force have a component along the motion?"W = Fd cos θ: perpendicular means cos 90° = 0. No work.
Friction losses recoveredEnergy audits that balance after a skid"Which account did the friction energy land in?"Heat. It never returns to the mechanical books. Write it as a loss term.
Energy and power interchangedMotors sized in joules, batteries in watts"Is this an amount or a rate?"Energy (J) is the amount; power (W = J/s) is the rate. P = Fv links them.
KE linear in speed"Twice the speed, twice the energy""What power of v sits in ½mv²?"Squared. Double speed means quadruple energy and quadruple braking distance.
06

Practice ladder

Level 1 · Direct skill

A hoist lifts a 200 kg pallet 6 m at constant speed in 8 s. Find the work done and the power required.

Show answer

W = mgh = 200 × 9.81 × 6 = 11.77 kJ. P = 11 772/8 = 1.47 kW. Add 20 to 30% for a real motor's losses.

Level 2 · Mixed concept

A 2 kg slider is released from rest at the top of a frictionless ramp 1.8 m high. Find its speed at the bottom, and explain why the ramp angle does not matter.

Show answer

mgh = ½mv²: v = √(2 × 9.81 × 1.8) = 5.94 m/s. Gravity's work depends only on the height drop, so every frictionless path from that height gives the same speed.

Level 3 · Independent problem

The Chapter 4 sled (net force 48.1 N on 20 kg) starts from rest. Use energy methods to find its speed after 10 m, and check with kinematics.

Show answer

Net work = 48.1 × 10 = 481 J = ½ × 20 × v², so v = √48.1 = 6.94 m/s. Kinematics: v = √(2 × 2.41 × 10) = 6.94 m/s. The theorem is Newton plus kinematics, pre-integrated.

Level 4 · Transfer to real engineering

Size a motor for a real task you choose (a winch, a conveyor lift, an e-bike hill climb): state mass, height or force, time target, and efficiency, and produce the required power with a margin.

What good work looks like

A clean energy audit, P = W/t with units, an efficiency assumption cited, and a sensible standard motor size chosen above the computed need.

07

Working with AI, and proving it yourself

Use AI as an examiner, not a solver

"Here is my energy audit with accounts and transfers. Find the account I missed or double-counted; do not rebalance it for me."
"Give me four scenarios; I will declare for each whether energy or force methods are cheaper, with one reason."
"How fast at the bottom?" The audit setup is the skill, not the arithmetic.
"What motor do I need?" Sizing with margins is engineering judgment to practice, not delegate.

Portfolio task

Write a one-page energy audit of a real system you use (your commute, a kettle, a gym session): all accounts, all transfers, the losses named, and one efficiency number computed from your own estimates.

Must include: a transfer diagram like the concept figure, at least one P = W/t computation, and a stated dominant loss.
08

Retrieval and spaced review

Closed notes. Answer out loud, then reveal.

1. Define work, with the angle dependence.

W = Fd cos θ: force times displacement times alignment. Perpendicular force does none.

2. State the work-energy theorem.

Net work on a body equals its change in kinetic energy: ΣW = Δ(½mv²).

3. Write the two potential energies of mechanics.

Gravitational mgh, spring ½ks². Both are stored work, recoverable.

4. What makes friction special in energy accounting?

Its work converts mechanical energy to heat irreversibly: a one-way leak, the seed of the second law in Thermodynamics.

5. Two ways to compute power?

P = W/t for averages and P = F·v instantaneously: the second sizes drives at speed.

TodayFinish this quiz and Levels 1 and 2 of the ladder.
+1 dayRe-solve the braking example both ways from memory.
+3 daysOne conservation problem with a spring in it.
+7 daysMixed set: an energy audit plus a Chapter 4 FBD.
+30 daysCarry the energy-accounts picture into Chapter 12's first law.
09

Textbook mapping

ItemMapping
Main sourceOpenStax University Physics Vol. 1, Work and Kinetic Energy and Potential Energy and Conservation of Energy
Benchmark / referenceMIT 8.01 · Young and Freedman
Core topics6.1 Constant-force work · 6.2 Varying-force work · 6.3 Work-energy theorem · 6.4 Potential energy · 6.5 Conservation · 6.6 Friction losses · 6.7 Power · 6.8 Efficiency
Engineering connectionDynamics, Thermodynamics (energy balance), Machine and motor sizing, Energy Systems.
Read nextChapter 7: Momentum, Impulse, and Collisions.