Unit 06 - Grade 11-12 Physics

Work, Energy, and Power

Learn how forces transfer energy, how energy changes form, how conservation laws simplify motion problems, and how power measures the rate of energy transfer.

Lesson roadmap

What Students Should Master in This Unit

Work and energy give students a powerful alternative to force-by-force analysis. Instead of tracking every moment of motion, energy methods compare initial and final states.

Calculate work

Use force, displacement, angle, and graphs to determine energy transfer.

Track energy

Connect kinetic, gravitational potential, elastic potential, thermal, and mechanical energy.

Apply conservation

Solve motion problems using work-energy and conservation of mechanical energy.

Quick navigation

Jump to a Topic

Work Work from Graphs Kinetic Energy Potential Energy Elastic Energy Work-Energy Theorem Conservation of Energy Nonconservative Work Power and Efficiency PhET Simulation Labs Worked Examples Practice Problems
Energy transfer

1. Work

In physics, work happens when a force causes displacement. Work is a transfer of energy into or out of a system.

Work by an angled force Only the force component along displacement does work displacement d F θ W = Fd cos(θ)
Work depends on the part of the force that points in the same direction as displacement.
Work by a constant force W = Fd cos(θ) θ is the angle between force and displacement.
Work unit 1 J = 1 N·m A joule is a newton-meter.
Positive work θ < 90 degrees Force component is in the direction of motion.
Negative work θ > 90 degrees Force component opposes motion.
Zero work θ = 90 degrees Force is perpendicular to displacement.
Net work Wnet = ΣW Add work done by all forces.
Common mistake: A force can be large and still do zero work if there is no displacement or if the force is perpendicular to displacement.
Area under force graph

2. Work from Force-Displacement Graphs

When force changes with position, work is found from the area under a force-displacement graph.

Work from graph area Area under an F vs. d graph equals work F d work = area constant force gives a rectangle
For constant force, the area is a rectangle. For changing force, use the shape under the curve.
Graph work W = area under F vs. d graph Use geometric area for simple shapes.
Rectangle area W = Fd Constant force.
Triangle area W = (1/2)base·height Useful for springs and linearly changing forces.

Spring Graph Connection

For a spring, the force increases with stretch or compression. The area under the force-position graph becomes elastic potential energy.

Es = (1/2)kx2
Energy of motion

3. Kinetic Energy

Kinetic energy is energy due to motion. Any moving object has kinetic energy.

Kinetic energy and speed Kinetic energy increases with the square of speed v 2v K 4K K = (1/2)mv2
Doubling speed makes kinetic energy four times larger because speed is squared.
Kinetic energy K = (1/2)mv2 Mass in kg, speed in m/s, energy in J.
Change in kinetic energy ΔK = Kf - Ki Final kinetic energy minus initial kinetic energy.
Speed from kinetic energy v = √(2K / m) Use when energy and mass are known.
Important: Kinetic energy depends on speed squared. Doubling speed makes kinetic energy four times larger.
Stored energy

4. Gravitational Potential Energy

Gravitational potential energy is energy stored because of height in a gravitational field. Near Earth's surface, it depends on mass, gravitational field strength, and height relative to a chosen zero level.

Gravitational potential energy Height above a chosen zero level stores gravitational energy h Fg zero height reference Ug = mgh
Only changes in height matter in most problems, so choose a zero level that makes the calculation simple.
Gravitational potential energy Ug = mgh Works near Earth's surface when g is constant.
Change in gravitational potential energy ΔUg = mgΔh Depends on change in height.
Work by gravity Wg = -ΔUg Gravity does positive work when an object moves downward.

Choosing Zero Height

The zero level for gravitational potential energy is a choice. Only changes in potential energy affect motion in most Grade 11-12 problems.

Springs

5. Elastic Potential Energy

Elastic potential energy is stored when a spring or elastic object is stretched or compressed.

Elastic potential energy A compressed or stretched spring stores elastic energy x release Us = (1/2)kx2
Spring energy depends on compression or stretch squared, so small changes in x matter a lot.
Hooke's law Fs = kx k is spring constant, x is stretch or compression.
Elastic potential energy Us = (1/2)kx2 Energy stored in a spring.
Spring constant unit N/m Stiffer springs have larger k values.
Common mistake: Spring energy uses x squared. Compressing a spring twice as far stores four times as much energy.
Net work changes motion

6. Work-Energy Theorem

The work-energy theorem states that net work equals the change in kinetic energy.

Work-energy theorem Net work changes kinetic energy Ki Kf Wnet Wnet = ΔK
If net work is positive, kinetic energy increases. If net work is negative, kinetic energy decreases.
Wnet = ΔK = Kf - Ki

When to Use It

  • When the problem asks for speed after a force acts over a distance.
  • When acceleration is not constant or time is not given.
  • When forces and displacement are easier to compare than forces and acceleration.
  • When a force-displacement graph is provided.
Connection: Newton's laws focus on force and acceleration. Energy methods focus on force and displacement.
Energy accounting

7. Conservation of Mechanical Energy

Mechanical energy is the sum of kinetic energy and potential energy. If only conservative forces do work, mechanical energy is conserved.

Conservation of mechanical energy Mechanical energy trades between potential and kinetic forms high Ug high K K + U stays constant if friction and air resistance are ignored
Without nonconservative work, total mechanical energy stays constant while energy changes form.
Mechanical energy Emech = K + Ug + Us Total mechanical energy.
Conservation statement Ei = Ef Use when no nonconservative work is involved.
Expanded form Ki + Ui = Kf + Uf Use for many ramps, falls, and roller-coaster problems.

Conservative Forces

Gravity and ideal spring forces are conservative. They can store and return mechanical energy.

Energy loss and transfer

8. Nonconservative Work

Nonconservative forces transfer mechanical energy into other forms such as thermal energy, sound, or internal energy. Friction is the most common example.

Friction and thermal energy Friction transforms mechanical energy into thermal energy motion friction thermal energy Wf = -fkd
Friction reduces mechanical energy, but total energy is still conserved because energy becomes thermal energy.
Nonconservative work Wnc = ΔEmech Mechanical energy changes when nonconservative work is done.
Work by kinetic friction Wf = -fkd Negative because friction usually opposes motion.
Energy with friction Ei + Wnc = Ef Useful when mechanical energy is not conserved.
Language note: Energy is not destroyed by friction. Mechanical energy is transformed into thermal energy and other forms.
Rate of energy transfer

9. Power and Efficiency

Power measures how quickly work is done or energy is transferred. Efficiency compares useful output energy or power to total input.

Power and efficiency Power measures how fast energy is transferred same work longer time means lower power same work shorter time means higher power P = W/t efficiency = useful output/input
A powerful machine or person can transfer the same energy in less time, while efficiency measures how much input becomes useful output.
Average power P = W / t Work divided by time.
Power from energy P = ΔE / t Energy transferred per second.
Power at constant velocity P = Fv Use when force and velocity are in the same direction.
Watt 1 W = 1 J/s A watt is one joule per second.
Efficiency efficiency = (useful output / total input) × 100% Can use energy or power.
Horsepower connection 1 hp ≈ 746 W Useful in real-world power comparisons.
Simulation labs

10. PhET Simulation Labs for This Unit

These official PhET simulations give students a hands-on way to test conservation of energy, work, springs, friction, energy transfer, and power-related ideas before or after tutoring sessions.

Simulation lab pathways Use simulations to test energy ideas before the lab report Skate Park K + U + thermal Springs elastic energy Ramp height and work evidence table adjust variables, observe energy bars, compare predictions
Simulation work becomes stronger when students record variables, energy changes, and evidence before writing conclusions.
Energy Skate Park

Explore kinetic energy, gravitational potential energy, thermal energy, friction, and conservation of energy using a skater on different tracks.

Lab idea: compare total energy with and without friction.
Open PhET Lab
Energy Forms and Changes

Study how energy is stored, transferred, and transformed between systems such as heat, light, mechanical motion, and electrical energy.

Lab idea: track energy pathways through a system.
Open PhET Lab
Masses and Springs

Use springs and masses to connect Hooke's law, elastic potential energy, kinetic energy, and gravitational potential energy.

Lab idea: change mass and spring constant, then observe energy exchange.
Open PhET Lab
The Ramp

Investigate force, work, energy, friction, and motion on a ramp with adjustable conditions.

Lab idea: compare work done on different ramp angles.
Open PhET Lab
Friction

Connect microscopic surface interactions to heating and energy transfer from mechanical energy to thermal energy.

Lab idea: explain where mechanical energy goes when friction acts.
Open PhET Lab
Forces and Motion: Basics

Review applied forces, friction, speed changes, and energy-related motion ideas from an earlier dynamics perspective.

Lab idea: connect force, distance, and speed changes.
Open PhET Lab
Investigation skills

11. Work, Energy, and Power Lab Skills

Labs in this unit usually involve measuring height, speed, force, distance, time, and energy transfer. Students should learn to compare predicted energy values with measured results.

Work and energy lab measurements Good energy labs connect measurements to predictions force sensor timer h v measure: m, h, v, F, d, t
Strong lab reports show measured values, calculated energy values, percent difference, and sources of uncertainty.

Common Lab Questions

  • How does height affect speed at the bottom of a ramp?
  • How does friction change mechanical energy?
  • How much work is done by an applied force?
  • How does spring compression affect launch speed?
  • How much power does a student generate climbing stairs?

Useful Lab Measurements

  • Mass in kilograms.
  • Height change in meters.
  • Speed in meters per second.
  • Force in newtons.
  • Distance in meters.
  • Time in seconds.
Worked examples

12. Worked Examples

Example 1: Work by a horizontal force

A 40.0 N force pushes a box 6.0 m in the same direction as motion. Find the work done.

W = Fd cos(θ) = (40.0)(6.0)cos(0 degrees) = 240 J.

Example 2: Work by an angled force

A 75 N force pulls a sled 10.0 m at 30.0 degrees above the horizontal. Find the work done by the force.

W = Fd cos(θ) = (75)(10.0)cos(30.0) = 650 J.

Example 3: Kinetic energy

A 2.0 kg cart moves at 5.0 m/s. Find kinetic energy.

K = (1/2)mv2 = (1/2)(2.0)(5.0)2 = 25 J.

Example 4: Gravitational potential energy

A 3.0 kg object is raised 4.0 m. Find the increase in gravitational potential energy.

ΔUg = mgΔh = (3.0)(9.8)(4.0) = 118 J.

Example 5: Conservation of energy

A 1.5 kg ball is dropped from rest from a height of 8.0 m. Ignore air resistance. Find speed just before impact.

mgh = (1/2)mv2. Mass cancels.

v = √(2gh) = √[2(9.8)(8.0)] = 12.5 m/s.

Example 6: Power climbing stairs

A 60.0 kg student climbs 4.0 m in 5.0 s. Find average power output against gravity.

Work against gravity = mgh = (60.0)(9.8)(4.0) = 2352 J.

P = W/t = 2352/5.0 = 470 W.

Independent practice

13. Practice Problems

Try each problem first. Then open the answer check and compare the formula, substitution, units, and reasoning.

1. A 25 N force moves an object 8.0 m in the direction of the force. Find work.

Answer

W = Fd = (25)(8.0) = 200 J.

2. A 50 N force acts at 60 degrees to the displacement over 4.0 m. Find work.

Answer

W = Fd cos(θ) = (50)(4.0)cos(60) = 100 J.

3. A force is perpendicular to displacement. How much work does it do?

Answer

Zero work, because cos(90 degrees) = 0.

4. Find the kinetic energy of a 4.0 kg object moving at 3.0 m/s.

Answer

K = (1/2)(4.0)(3.0)2 = 18 J.

5. A 2.0 kg object has 100 J of kinetic energy. Find speed.

Answer

v = √(2K/m) = √[2(100)/2.0] = 10 m/s.

6. A 5.0 kg object is lifted 2.0 m. Find gravitational potential energy gained.

Answer

ΔUg = mgΔh = (5.0)(9.8)(2.0) = 98 J.

7. A spring with k = 200 N/m is compressed 0.10 m. Find elastic potential energy.

Answer

Us = (1/2)kx2 = (1/2)(200)(0.10)2 = 1.0 J.

8. Net work of 300 J is done on a cart. Its initial kinetic energy was 100 J. Find final kinetic energy.

Answer

Wnet = ΔK, so Kf = 100 + 300 = 400 J.

9. A 2.0 kg ball falls from 5.0 m. Ignore air resistance. Find speed before impact.

Answer

mgh = (1/2)mv2, so v = √(2gh) = √[2(9.8)(5.0)] = 9.9 m/s.

10. A friction force of 12 N acts over 5.0 m opposite motion. Find work by friction.

Answer

Wf = -fd = -(12)(5.0) = -60 J.

11. A machine does 1200 J of work in 6.0 s. Find power.

Answer

P = W/t = 1200/6.0 = 200 W.

12. A person outputs 400 W for 10 s. How much energy is transferred?

Answer

E = Pt = (400)(10) = 4000 J.

13. A motor uses 1000 J of input energy and produces 750 J of useful output. Find efficiency.

Answer

Efficiency = (750/1000)(100%) = 75%.

14. A force-displacement graph is a triangle with base 6.0 m and height 20 N. Find work.

Answer

Area = (1/2)(6.0)(20) = 60 J.

15. A 1200 kg car moving at 20 m/s doubles its speed to 40 m/s. By what factor does kinetic energy change?

Answer

Kinetic energy depends on v2, so it increases by a factor of 4.

16. In Energy Skate Park, friction is turned on. What happens to mechanical energy and total energy?

Answer

Mechanical energy decreases as some energy becomes thermal, but total energy remains conserved.

Final review

14. What to Know Before Moving On

  • Work is energy transferred by a force over displacement.
  • W = Fd cos(θ) for constant force.
  • Area under a force-displacement graph gives work.
  • Kinetic energy is K = (1/2)mv2.
  • Gravitational potential energy near Earth is Ug = mgh.
  • Elastic potential energy is Us = (1/2)kx2.
  • Net work changes kinetic energy: Wnet = ΔK.
  • Mechanical energy is conserved when only conservative forces do work.
  • Friction transforms mechanical energy into thermal energy.
  • Power is the rate of energy transfer: P = W/t.
Previous Unit: Complex Force Systems Back to Unit Library Next Unit: Momentum and Collisions

Useful Links

Physics Units

Support

© 2025 Physicstutors.org | All rights reserved