Work, Energy, and Power

Big Picture

Energy is the currency of physics. You can convert it between forms — kinetic, potential, thermal — but you cannot create or destroy it. Work is the mechanism by which forces transfer energy into or out of a system. Power describes how fast that transfer happens.

ELI-10: Explain It Like I'm 10

Energy is like money in a bank account. Kinetic energy is the money you have "in motion." Potential energy is money stored up, ready to be spent. Work is like making a deposit or withdrawal — a force moving something transfers energy. Power is how fast you are spending or earning. You can move money between accounts, but the total amount stays the same (conservation of energy).

3.1 Work

Core Idea

Work is done when a force acts on an object and the object moves in the direction of the force (or has a component in that direction).

Physics and Mathematics

For a constant force: W = F d cosθ= F · d

Where:

• W = work (joules, J; 1 J = 1 N·m)
• F = magnitude of the force (N)
• d = magnitude of displacement (m)
• θ = angle between the force and displacement vectors

Key cases:

• θ= 0^° → W = Fd (force in direction of motion, maximum positive work)
• θ= 90^° → W = 0 (force perpendicular to motion; e.g., normal force on a horizontal surface does zero work)
• θ= 180^° → W = -Fd (force opposes motion; e.g., friction does negative work)

For a variable force, work is the area under the force-versus-position curve: W = ∫F(x) dx. On an F-vs-x graph, count the area between the curve and the x-axis from initial to final position. Area above the axis is positive work; below is negative.

Graphically: If you push a spring, the force increases linearly with compression (F = kx). The area under the F-vs-x line from x = 0 to x = d is a triangle: W = 12(kd)(d) = 12kd², which equals the elastic potential energy stored.

Net work is the sum of work done by all forces, or equivalently the work done by the net force.

Conceptual Example

You push a box with 50 N horizontally for 3.0 m. The work you do is W = (50)(3.0)cos0^°= 150 J. Friction pushes back with 20 N and does W = (20)(3.0)cos180^°= -60 J. Net work = 90 J.

ELI-10: Explain It Like I'm 10

Carrying a heavy box across the room while holding it level? You might feel tired, but physically you did zero work on the box — because your force (upward) is perpendicular to the motion (sideways). Work requires force and motion in the same direction. If you lift the box straight up, now you are doing work.

Common Mistakes

• "Holding a heavy object stationary requires work." No — zero displacement means zero work, regardless of how tired you feel. Your muscles do internal work, but no physical work is done on the object.
• Confusing positive and negative work. Work can be negative — that means energy is being removed from the object.

Key Takeaway

Work = (force component along displacement) × (displacement). Zero displacement or perpendicular force → zero work.

3.2 Kinetic Energy and the Work-Energy Theorem

Core Idea

Kinetic energy is the energy of motion. The net work done on an object equals its change in kinetic energy.

Physics and Mathematics

K = 12mv²

Where:

• K = kinetic energy (J)
• m = mass (kg)
• v = speed (m/s)

Work-energy theorem: W_net = ΔK = K_f - K_i = 12mv_f² - 12mv_i²

Why v²? Doubling the speed quadruples the kinetic energy. This is why a car at 60 mph has four times the kinetic energy of the same car at 30 mph — and thus four times the stopping distance.

Worked Example: Car Braking

Problem: A 1200 kg car traveling at 25 m/s brakes to a stop. How much work is done by the brakes?

Solution: W_net = ΔK = 0 - 12(1200)(25)² = -375,000 J

Answer: −375 kJ. The negative sign means energy is removed from the car (converted to heat in the brakes).

ELI-10: Explain It Like I'm 10

Kinetic energy is the energy something has just because it is moving. A rolling bowling ball has more kinetic energy than a rolling marble at the same speed because it is heavier. If you double the speed, you do not double the energy — you quadruple it. That is why high-speed crashes are so much worse than low-speed ones.

Common Mistakes

• Using mv² instead of 12mv².
• Thinking momentum (mv) and kinetic energy (12mv²) are the same. They are different quantities (Topic 4 clarifies this).

Key Takeaway

K = 12mv². Net work equals the change in kinetic energy. Negative net work means the object slows down.

3.3 Potential Energy

Core Idea

Potential energy is stored energy that depends on an object's position or configuration. It represents the potential to do work.

Gravitational Potential Energy

Near Earth's surface: U_g = mgh

Where:

• U_g = gravitational potential energy (J)
• m = mass (kg)
• g = 9.8 m/s²
• h = height above a chosen reference level (m)

The reference level (h = 0) is arbitrary — only changes in potential energy are physically meaningful.

Elastic Potential Energy (Springs)

For an ideal spring obeying Hooke's law (F = -kx): U_s = 12kx²

Where:

• U_s = elastic potential energy (J)
• k = spring constant (N/m), a measure of stiffness
• x = displacement from equilibrium (m)

ELI-10: Explain It Like I'm 10

Hold a ball above the ground. It is not moving, so it has no kinetic energy. But drop it, and it speeds up — energy appeared from somewhere. That "somewhere" is gravitational potential energy: the ball's height gives it stored energy, which converts to motion-energy as it falls. A stretched rubber band or a compressed spring is the same idea — you put energy in by stretching it, and it releases that energy when you let go.

Proportional Reasoning for Energy

• Kinetic energy: K ∝ m (double mass → double K). K ∝ v² (double speed → quadruple K). This is the most important non-linear proportionality in introductory physics.
• Gravitational PE: U_g ∝ m and U_g ∝ h. Double either → double stored energy.
• Spring PE: U_s ∝ k and U_s ∝ x². Double the compression → quadruple the stored energy. This is why a spring compressed twice as far stores four times the energy, and why compressing a stiff spring is much harder than a soft one.

Worked Example: Spring Compression

Problem: A spring with k = 500 N/m is compressed by 0.10 m. How much energy is stored? If this energy is transferred entirely to a 0.050 kg ball, what is the ball's launch speed?

Given: k = 500 N/m, x = 0.10 m, m = 0.050 kg.

Solution: U_s = 12kx² = 12(500)(0.10)² = 2.5 J. Then K = U_s = 12mv² → v = √2K/m = √2(2.5)/0.050 = 10 m/s.

Does it Make Sense? 10 m/s (about 22 mph) from a compressed spring is reasonable — think of a pinball launcher. If you double the compression to 0.20 m, U_s quadruples to 10 J and launch speed doubles to 20 m/s. This square relationship means small changes in compression produce large changes in stored energy.

Model Assumptions

The energy equations in this topic assume:

• Point mass: object's size and shape do not matter for energy calculations.
• Ideal spring: obeys Hooke's law exactly (F = -kx), no internal friction, massless.
• Constant g: near Earth's surface, g = 9.8 m/s². Not valid for satellites or interplanetary distances.
• No air resistance: projectile and pendulum energy conservation analyses ignore drag.

Common Mistakes

• Treating h as an absolute height. Only Δh matters. Choose a convenient reference level.
• Confusing gravitational potential energy near Earth (mgh) with the universal form (-GMm/r, Topic 6).

Key Takeaway

Potential energy is stored by position (gravity) or configuration (springs). Only changes in potential energy are physical.

3.4 Conservation of Mechanical Energy

Core Idea

In a system with only conservative forces (gravity, ideal springs), the total mechanical energy is constant: E = K + U = constant

Conservative vs Non-Conservative Forces

• Conservative forces: Work done is independent of path; mechanical energy is conserved. Examples: gravity, ideal spring force.
• Non-conservative forces: Work depends on path; mechanical energy is NOT conserved (some converts to thermal energy). Examples: friction, air resistance, applied forces.

General energy equation: K_i + U_i + W_nc = K_f + U_f

Where W_nc is work done by non-conservative forces. Friction makes W_nc negative.

Worked Example: Roller Coaster

Problem: A roller coaster starts from rest at height 30 m. Neglecting friction, what is its speed at ground level?

Solution: K_i + U_i = K_f + U_f 0 + mgh = 12mv² + 0 Cancel m: gh = 12v² v = √2gh = √2(9.8)(30) ≈ 24.2 m/s

Key insight: The mass canceled. All objects — regardless of mass — reach the same speed when dropped from the same height (assuming no friction). The answer (~54 mph) is physically reasonable for a coaster drop.

ELI-10: Explain It Like I'm 10

A roller coaster at the top of a hill has lots of stored (potential) energy and no motion energy. As it rolls down, stored energy turns into speed. At the bottom, all the stored energy has become motion energy. On the way up the next hill, motion energy turns back into stored energy. Without friction, this would go on forever — the total never changes. Friction slowly leaks energy out as heat, which is why real coasters need chain lifts.

3.5 Power

Core Idea

Power is the rate at which work is done or energy is transferred.

Physics and Mathematics

Average power: P_avg = WΔt

Instantaneous power (force and velocity): P = F · v = Fvcosθ

Where:

• P = power (watts, W; 1 W = 1 J/s)
• W = work (J)
• Δt = time interval (s)

Conceptual Example

Two people each lift a 50 kg crate 2 m (W = mgh ≈ 980 J each). One takes 2 seconds (P = 490 W); the other takes 10 seconds (P = 98 W). Same work, different power — the faster lifter is more powerful.

ELI-10: Explain It Like I'm 10

Work is the total amount of effort. Power is how fast you do that effort. Walking up a flight of stairs takes the same total work whether you sprint or stroll — but sprinting requires more power because you do the same work in less time. A lightbulb's wattage tells you how fast it converts electrical energy into light and heat.

Key Takeaway

Power = rate of energy transfer. P = W/Δt. A watt is a joule per second.

Limiting Cases for Energy

• What if friction vanishes? Mechanical energy is perfectly conserved. A pendulum swings forever. No real system achieves this, but it is a useful starting approximation.
• What if v → 0? Kinetic energy goes to zero. All the object's mechanical energy is now potential (if at height) or stored in springs.
• What if h → 0? Gravitational PE goes to zero. On level ground, energy exchanges only between kinetic and spring/thermal forms.
• What if the spring constant k → ∞? The spring becomes an effectively rigid rod — no energy can be stored, and the system behaves like an inelastic collision (see Topic 4).

Topic Summary

• Work = Fdcosθ. Force must have a component along the displacement.
• Kinetic energy K = 12mv². Net work = ΔK.
• Gravitational PE U_g = mgh. Elastic PE U_s = 12kx².
• Conservation of mechanical energy holds when only conservative forces act.
• Non-conservative forces (friction) remove mechanical energy, converting it to thermal energy.
• Power = rate of energy transfer, P = W/Δt = Fvcosθ.

Essential Equations

Concept Check

1. A person carries a suitcase horizontally at constant speed. How much work does the person do on the suitcase? Explain.
2. A spring is compressed by 2 cm, then by 4 cm. How does the stored energy in the second case compare to the first?
3. A satellite orbits Earth in a circular path. Does gravity do work on the satellite? (Hint: direction of force vs. direction of motion.)
4. If you double the speed of a car, by what factor does its kinetic energy increase? Its stopping distance?
5. Two ramps — one steep, one gradual — reach the same height. Neglecting friction, which requires more work to push a box to the top?

Open Educational References

• OpenStax, College Physics, Chapter 7: Work, Energy, and Energy Resources
• OpenStax, University Physics, Volume 1, Chapters 7–8: Work and Energy, Potential Energy and Conservation