Work, Energy and Power
Forces cause changes in motion, but analyzing motion using forces alone is laborious.
Start with the simplest version: this lesson is about Work, Energy and Power. If you can explain the core idea to a friend using everyday language, examples, and one clear reason why it matters, you have moved from memorising to understanding.
Forces cause changes in motion, but analyzing motion using forces alone is laborious. Energy provides an elegant alternative: instead of tracking every force, we track energy transformations. When a roller coaster climbs, gravitational potential energy increases; as it descends, this converts to kinetic energy. Understanding work, energy, and power reveals the deep unity underlying all physical processes.
Work: Force Doing Its Job
In everyday language, "work" means any effort or task. In physics, work has a precise definition: W = F · s · cos(θ), where F is force magnitude, s is displacement magnitude, and θ is the angle between them.
Key insight: Only the component of force in the direction of motion does work. If you push horizontally on a box and it slides horizontally, you do positive work. If you push at an angle, only the horizontal component counts. If you push perpendicular to motion (like supporting a book while walking horizontally), you do zero work, though you're "working" in the everyday sense.
Work is a scalar (just a number, no direction). It can be positive (force aids motion), negative (force opposes motion), or zero (force perpendicular to motion). Units: joules (J), where 1 J = 1 newton-meter.
A powerful thought experiment: If you push a box across the floor and it slides 10 meters, you do positive work. If you push up while holding the box and walking sideways, the upward force does zero work (perpendicular to motion). Gravity does negative work as you carry it upstairs (opposes upward motion).
Kinetic Energy: Energy of Motion
An object in motion can do work. A moving hammer can drive a nail; a speeding car can crush an obstacle. This capacity to do work is kinetic energy: KE = ½mv².
Kinetic energy depends on mass (heavier objects have more) and velocity squared (doubling speed quadruples energy). This velocity-squared relationship is crucial: it explains why car crashes at 60 mph are much more dangerous than at 30 mph.
Where does kinetic energy come from? Forces do work on the object, increasing its speed. The work-energy theorem states: Net work equals change in kinetic energy: W_net = ΔKE = ½mv_f² − ½mv_i².
This is profoundly useful. Instead of analyzing forces to find velocity, calculate work and use the work-energy theorem directly.
The Work-Energy Theorem
Imagine a car accelerating on a straight road. Calculating the final velocity using kinematics (v² = u² + 2as) requires knowing acceleration. But acceleration depends on the net force, which depends on detailed force analysis.
Alternatively: Calculate the work done by the engine and resistance (friction, air drag). The net work equals the change in kinetic energy. This often requires less detailed force information.
The theorem applies to any situation: a thrown ball slowing down due to air resistance, an object sliding to a stop due to friction, or a projectile under gravity. Work done by all forces equals kinetic energy change.
Potential Energy: Stored Energy
Lift a ball upward. You do work against gravity. The ball gains the capacity to do work later—if you release it, gravity pulls it downward, and it accelerates. This stored capacity is potential energy: PE_gravity = mgh, where h is height.
Potential energy depends on position. The higher an object, the more gravitational potential energy. Importantly, potential energy is relative—we choose a reference height (like the ground) as our zero point. What matters is differences in potential energy, not absolute values.
Another form: elastic potential energy in a stretched spring, PE_spring = ½kx², where k is spring stiffness and x is stretch. Compressed springs and stretched rubber bands store potential energy.
Potential energy is associated with conservative forces—forces that depend only on position, not on how you got there. Gravitational and elastic forces are conservative. Non-conservative forces (like friction) dissipate energy, converting it to heat.
Conservation of Mechanical Energy
Here's the beautiful insight: in a system with no friction or non-conservative forces, total mechanical energy is conserved.
Mechanical energy = kinetic energy + potential energy = constant (if only conservative forces act).
Imagine a pendulum swinging. At the peak, velocity is zero (KE = 0), but height is maximum (PE = maximum). Total energy is PE_max + 0. At the bottom, height is zero (PE = 0), but velocity is maximum (KE = maximum). Total energy is 0 + KE_max. These equal each other!
Energy continuously transforms between forms, but the total remains constant. This conservation law is powerful: instead of analyzing forces and acceleration, track energy transformations.
In real systems, friction acts and dissipates energy as heat. "Dissipation" doesn't violate energy conservation—it converts mechanical energy to thermal energy. Total energy (including heat) is always conserved, a principle central to thermodynamics.
Power: Energy per Unit Time
Power is the rate of doing work: P = W/t, measured in watts (W), where 1 W = 1 joule/second.
A light bulb rated 100 W uses 100 joules of energy per second. A car engine rated 100 kilowatts produces 100,000 joules of work per second. Two engines doing the same work in different times have different power—the faster one has higher power.
Power is crucial in engineering and biology. Your muscles have limited power output (why you can't jump as high as a kangaroo). A sprinter produces enormous power for short duration; a marathoner produces less power but sustains it longer.
Real-World Applications
Understanding energy explains technologies and natural phenomena:
- Why roller coasters work: gravitational potential energy converts to kinetic energy on descent
- Why brakes overheat: kinetic energy is dissipated as heat by friction
- Why electric cars are efficient: less wasted energy compared to burning fuel
- How hydropower works: falling water (gravitational PE) drives turbines
Socratic Questions
- You push a box across the floor. The box accelerates, moves at constant velocity, and then decelerates when you stop pushing. Using the work-energy theorem, explain what's happening to energy in each phase.
- A ball thrown upward slows down as it rises. Where does its kinetic energy go? What form does it take at the highest point?
- Why does doubling your speed have a much bigger effect on kinetic energy than doubling your mass? What does the velocity-squared relationship tell us about collision safety?
- A roller coaster climbs a 50-meter hill, starting from rest. How fast is it moving at the bottom, assuming no friction? (Hint: use energy conservation with PE = mgh.)
- Why can't a single elastic collision between objects increase their total kinetic energy? How does this relate to conservative forces?