What you'll learn
This topic examines the fundamental relationship between energy and work in physical systems. Students must understand how forces cause energy transfers, calculate work done in various scenarios, and apply conservation of energy principles. These concepts appear in both calculation-based and conceptual questions across Common Core physics assessments.
Key terms and definitions
Work — energy transferred when a force moves an object through a distance; measured in joules (J). Work is only done when the force has a component in the direction of motion.
Energy — the capacity of a system to do work; exists in multiple forms including kinetic, gravitational potential, elastic potential, thermal, and chemical energy.
Kinetic energy — energy possessed by an object due to its motion, calculated using KE = ½mv², where m is mass (kg) and v is velocity (m/s).
Gravitational potential energy — energy stored in an object due to its position in a gravitational field, calculated using GPE = mgh, where m is mass (kg), g is gravitational field strength (9.8 m/s² on Earth), and h is height (m).
Power — the rate at which work is done or energy is transferred; measured in watts (W), where 1 watt = 1 joule per second.
Conservation of energy — the principle that energy cannot be created or destroyed, only transferred from one form to another or between objects.
Efficiency — the ratio of useful energy output to total energy input, often expressed as a percentage.
Elastic potential energy — energy stored when an object is stretched, compressed, or deformed, provided it returns to its original shape.
Core concepts
Calculating work done
Work is calculated using the equation:
W = Fd cos θ
Where:
- W = work done (J)
- F = force applied (N)
- d = displacement (m)
- θ = angle between force and displacement
When force and displacement are in the same direction, θ = 0° and cos θ = 1, simplifying to:
W = Fd
Critical principles:
- Work is a scalar quantity despite involving vector quantities (force and displacement)
- Only the component of force parallel to displacement does work
- If force is perpendicular to motion (θ = 90°), no work is done
- Negative work occurs when force opposes motion (θ = 180°)
Common scenarios in US Common Core physics assessments:
- Lifting objects vertically against gravity: W = mgh
- Pushing objects horizontally: W = Fd
- Forces at angles: requires cos θ calculation
- Work against friction: W = friction force × distance
Kinetic energy and the work-energy theorem
The work-energy theorem states that the net work done on an object equals its change in kinetic energy:
W_net = ΔKE = ½mv_f² - ½mv_i²
This fundamental relationship connects forces (which do work) to motion (which involves kinetic energy). Applications include:
- Calculating final velocity after work is done on an object
- Determining stopping distances when friction does negative work
- Analyzing collisions and energy transfers
For an object starting from rest, all work done becomes kinetic energy:
W = ½mv²
For an object brought to rest, the initial kinetic energy equals the work required:
W = -½mv_i² (negative because force opposes motion)
Gravitational potential energy
Objects gain gravitational potential energy when raised in a gravitational field. Near Earth's surface, with uniform gravitational field strength:
GPE = mgh
Key examination points:
- GPE is measured relative to a reference level (often ground level or the starting position)
- Raising an object requires work equal to the GPE gained: W = mgh
- The path taken doesn't matter—only the vertical height change
- GPE can be negative if the object is below the reference level
Conservation of energy in mechanical systems
In the absence of non-conservative forces (like friction and air resistance), mechanical energy remains constant:
E_initial = E_final
KE_i + GPE_i = KE_f + GPE_f
This principle solves numerous Common Core physics problems:
Falling objects: Object dropped from height h:
- Initially: KE = 0, GPE = mgh
- Just before impact: KE = ½mv², GPE = 0
- Therefore: mgh = ½mv²
- Solving: v = √(2gh)
Pendulums: Maximum KE at lowest point equals maximum GPE at highest point.
Roller coasters: Sum of KE and GPE remains constant (ignoring friction).
When friction or air resistance acts, mechanical energy decreases:
E_mechanical,initial = E_mechanical,final + E_thermal
The "lost" mechanical energy converts to thermal energy in the surroundings.
Elastic potential energy
When springs or elastic materials deform, they store elastic potential energy:
EPE = ½kx²
Where:
- k = spring constant (N/m)
- x = extension or compression from equilibrium position (m)
Common Core assessments frequently combine elastic and kinetic energy:
- Spring-launched projectiles: EPE converts to KE
- Collision with springs: KE converts to EPE
- Maximum compression occurs when all KE converts to EPE
Power calculations
Power quantifies the rate of energy transfer or work done:
P = W/t or P = E/t
Where:
- P = power (W)
- W = work (J)
- E = energy transferred (J)
- t = time (s)
Alternative power equation when constant force causes constant velocity:
P = Fv
Where:
- F = force (N)
- v = velocity (m/s)
This equation appears in problems involving vehicles traveling at constant speed against resistance forces.
Efficiency
No energy transfer is perfectly efficient. Efficiency is calculated:
Efficiency = (useful energy output / total energy input) × 100%
Or equivalently:
Efficiency = (useful power output / total power input) × 100%
Common contexts:
- Electric motors converting electrical energy to kinetic energy
- Inclined planes reducing force required (mechanical advantage vs. efficiency)
- Power stations converting fuel energy to electrical energy
The "wasted" energy typically dissipates as thermal energy due to friction, air resistance, or electrical resistance.
Worked examples
Example 1: Work and kinetic energy
Question: A 1200 kg car accelerates from rest. The engine applies a constant forward force of 3600 N over a distance of 50 m. Calculate:
a) The work done by the engine (2 marks)
b) The final velocity of the car, assuming no friction (3 marks)
Solution:
a) W = Fd
W = 3600 N × 50 m = 180,000 J (or 180 kJ)
Award 1 mark for correct equation, 1 mark for correct answer with unit
b) Using work-energy theorem:
W = ΔKE = ½mv² - 0 (starts from rest)
180,000 = ½ × 1200 × v²
180,000 = 600v²
v² = 300
v = 17.3 m/s
Award 1 mark for applying work-energy theorem, 1 mark for correct rearrangement, 1 mark for correct answer with unit
Example 2: Conservation of energy
Question: A 0.5 kg ball is dropped from a window 20 m above the ground. Calculate:
a) The gravitational potential energy lost as it falls (2 marks)
b) The speed just before it hits the ground, neglecting air resistance (3 marks)
Solution:
a) GPE = mgh
GPE = 0.5 kg × 9.8 m/s² × 20 m = 98 J
Award 1 mark for correct equation, 1 mark for correct answer
b) Using conservation of energy:
GPE_initial = KE_final
mgh = ½mv²
gh = ½v² (mass cancels)
9.8 × 20 = ½v²
196 = ½v²
v² = 392
v = 19.8 m/s
Award 1 mark for energy conservation principle, 1 mark for correct calculation, 1 mark for answer with unit. Alternative approach using v² = 2gh also acceptable.
Example 3: Power
Question: An electric motor lifts a 300 kg load through a vertical height of 12 m in 15 seconds. Calculate:
a) The work done against gravity (2 marks)
b) The minimum power output of the motor (2 marks)
Solution:
a) W = mgh
W = 300 kg × 9.8 m/s² × 12 m = 35,280 J
Award 1 mark for equation, 1 mark for answer
b) P = W/t
P = 35,280 J / 15 s = 2352 W (or 2.35 kW)
Award 1 mark for equation, 1 mark for answer with unit
Common mistakes and how to avoid them
Confusing work and energy units — Students write "watts" for work or "joules" for power. Always use joules (J) for work and energy; watts (W) for power. Remember: 1 W = 1 J/s.
Forgetting that force and displacement must be parallel for W = Fd — When force acts at an angle, use W = Fd cos θ. Carrying a suitcase horizontally involves zero work by the upward force because displacement is perpendicular to force.
Incorrect GPE reference points — GPE depends on the chosen zero level. If an object moves from height h₁ to h₂, the change is mg(h₂ - h₁), not mgh₂. Always clearly define the reference level.
Not accounting for energy dissipation — In real systems with friction, mechanical energy decreases. Don't assume conservation of mechanical energy unless told to "neglect friction" or "assume no air resistance."
Mass cancellation errors — When using mgh = ½mv² for falling objects, mass cancels to give v = √(2gh). Students sometimes incorrectly keep mass in the equation, leading to calculation errors.
Power equation misuse — P = Fv applies only when force and velocity are constant and parallel. For average power over changing conditions, use P = W/t or P = E/t.
Exam technique for Energy and Work
Identify the command words: "Calculate" requires numerical answer with units and working. "Explain" needs a causal chain using physics principles. "State" needs brief factual response without justification. "Derive" requires mathematical manipulation showing each step.
Show all working for calculations: Even if the final answer is incorrect, intermediate steps earn method marks. Write equation first, substitute values, then calculate. Always include units in final answers—omitting units typically loses the final mark.
Energy transfer chains: When questions ask about energy conversions, specify both the initial and final forms (e.g., "chemical energy in fuel → kinetic energy of vehicle"). Vague statements like "energy is used" score zero marks.
Sketch work done on force-displacement graphs: The area under a force-distance graph equals work done. Common Core assessments may present this graphically—recognize that triangular areas indicate changing force, while rectangular areas show constant force.
Quick revision summary
Work equals force times displacement in the force direction (W = Fd cos θ), measured in joules. Kinetic energy (½mv²) increases when net work is done on an object. Gravitational potential energy (mgh) depends on height above a reference level. Conservation of energy states total energy remains constant; in mechanical systems without friction, KE + GPE stays constant. Power (P = W/t) measures rate of energy transfer in watts. Efficiency compares useful energy output to total input as a percentage. All energy transfers ultimately dissipate some energy as thermal energy.