7.1 Work, Power, and Mechanical Energy
Key Takeaways
- Work in physics requires a force and a displacement in the direction of that force: holding a heavy box motionless or carrying it horizontally does no work on the box against gravity, no matter how tiring it feels.
- Power is the rate of doing work, P = W/t, measured in watts; two people who lift identical loads to the same height do equal work, but the faster one develops more power.
- Kinetic energy is KE = ½mv², so doubling speed quadruples kinetic energy — the reason vehicle stopping distance grows far faster than speed does.
- Gravitational potential energy is PE = mgh and depends on the height above a chosen reference level, so the reference must be stated before a PE value means anything.
- In the absence of friction, mechanical energy is conserved: KE + PE stays constant, so a pendulum has maximum PE and zero KE at the top of its swing and maximum KE with minimum PE at the bottom.
Precision About Three Familiar Words
Work, power, and energy all have loose everyday meanings and precise scientific ones, and Competency 009 is written to test the precise versions. Exam items routinely present a scenario that feels like hard work — a student straining to hold a heavy box — and ask how much work is done on the box. The answer is zero, and it depends entirely on the definition.
Work
Work is done when a force moves an object through a distance in the direction of that force.
W = F × d, measured in joules (J), where 1 J = 1 N·m.
Three cases where the answer is zero joules, all of them common item stems:
- No displacement. Pushing hard against a brick wall that does not move: d = 0, so W = 0.
- Force perpendicular to motion. Carrying a box horizontally at constant height: the upward supporting force is perpendicular to the horizontal displacement, so that force does no work on the box.
- No force in the direction of motion. A puck sliding on frictionless ice at constant velocity has no net force acting along its path.
Worked example. A custodian pushes a cart with 45 N of horizontal force across a 12 m hallway. W = 45 × 12 = 540 J. If the custodian instead lifts a 15 kg box from the floor to a 1.2 m shelf, the lifting force must at least equal the weight: F = mg = 15 × 9.8 = 147 N, so W = 147 × 1.2 = 176 J.
Power
Power is the rate at which work is done: P = W / t, measured in watts (W), where 1 W = 1 J/s.
Power separates two performances that involve identical work. If two students each carry a 200 N stack of books up a 3.0 m staircase, both do 600 J of work. The student who takes 5 s develops 600 ÷ 5 = 120 W; the one who takes 12 s develops 600 ÷ 12 = 50 W. Same work, different power.
Useful anchors for plausibility checks: a compact fluorescent lamp draws roughly 15 W, a microwave oven about 1,000 W, and a sustained human working effort is on the order of 75-100 W.
Kinetic Energy
KE = ½mv², in joules. Because velocity is squared, speed dominates:
| Mass | Speed | Kinetic energy |
|---|---|---|
| 1,000 kg | 10 m/s | ½(1,000)(100) = 50,000 J |
| 1,000 kg | 20 m/s | ½(1,000)(400) = 200,000 J |
| 2,000 kg | 10 m/s | ½(2,000)(100) = 100,000 J |
Doubling the mass doubles KE; doubling the speed quadruples it. This is the physics behind highway safety messaging and a reliable exam item: a car at 60 km/h carries four times the kinetic energy it had at 30 km/h, so the braking distance required to dissipate that energy is roughly four times as long.
Gravitational Potential Energy
PE = mgh, where g ≈ 9.8 m/s² and h is height above a stated reference level. A 2.0 kg textbook on a 0.75 m desk has PE = 2.0 × 9.8 × 0.75 = 14.7 J relative to the floor — but relative to the desktop its PE is zero. Insisting that students name the reference is not pedantry; it prevents the confusion that arises when a problem switches from floor level to ground level outside the window.
Elastic potential energy is the other form 4-8 students meet: energy stored in a stretched rubber band, a compressed spring, or a drawn bow. Like gravitational PE, it is energy of configuration rather than motion.
Conservation of Mechanical Energy
With friction and air resistance negligible, KE + PE remains constant.
Worked example — a ramp. A 0.5 kg cart is released from rest at the top of a frictionless 1.2 m ramp.
- At the top: PE = 0.5 × 9.8 × 1.2 = 5.88 J, KE = 0.
- At the bottom: PE = 0, so KE = 5.88 J.
- Solving ½mv² = 5.88 for v: v² = (2 × 5.88) ÷ 0.5 = 23.5, so v ≈ 4.85 m/s.
Notice the mass cancels if you set mgh = ½mv², giving v = √(2gh). Every object released from the same frictionless height arrives at the same speed regardless of mass — the same principle behind equal free-fall acceleration.
A pendulum makes the exchange visible. At the highest point of the swing the bob is momentarily at rest: PE is maximum, KE is zero. At the lowest point PE is minimum and KE is maximum, so the bob moves fastest. At intermediate points the energy is shared. In a real pendulum the swing decays because air resistance and pivot friction convert mechanical energy into thermal energy — the energy is not destroyed, and its total is still conserved, which is the bridge to the conservation laws developed later in this domain.
Distinguishing the Three Quantities
| Question | Quantity | Formula | Unit |
|---|---|---|---|
| How much energy was transferred by a force? | Work | W = Fd | joule |
| How fast was that transfer? | Power | P = W/t | watt |
| How much energy does motion carry? | Kinetic energy | KE = ½mv² | joule |
| How much energy does position store? | Potential energy | PE = mgh | joule |
Work and energy share the joule because work is a transfer of energy. Power alone measures a rate, which is why it alone carries a per-second unit.
A student holds a 60 N backpack motionless at shoulder height for 90 seconds and reports feeling exhausted. How much work is done on the backpack, and why?
Two elevators each raise a 4,000 N load to a height of 15 m. Elevator A takes 10 s and Elevator B takes 25 s. Which comparison is correct?
A 0.20 kg pendulum bob is pulled back so it rises 0.45 m above its lowest point and released. Ignoring friction, what is its speed at the lowest point?