Understanding
Energy
Energy is the capacity to do work. This guide builds from the definition of work through kinetic and potential energy, conservation principles (including the pendulum), and on to power and efficiency.
Work is how energy is transferred. Pushing a box horizontally while a force acts upward does no useful work — only the component along the direction of motion counts. This is why W = F·d·cosθ: cos θ extracts exactly the component of force that is parallel to the displacement.
Imagine pulling a suitcase at an angle. The upward component of your pull lifts the case slightly — it doesn't contribute to moving it forward. Only the horizontal component does work against friction and moves the suitcase along the floor. The perpendicular component merely changes the normal force.
Force is horizontal, displacement is horizontal. θ = 0°, so cos 0° = 1.
The normal force acts vertically upward. The displacement is horizontal. θ = 90°, cos 90° = 0.
b) A 60 N force is applied at 45° above horizontal. The object moves 5 m. What is the work done by the force?
c) You carry a box horizontally at constant height across a room. How much work does gravity do on the box?
a) W = 40 × 8 × cos 0° = 320 J
b) W = 60 × 5 × cos 45° = 300 × 0.707 = 212 J
c) Zero. Gravity acts downward; displacement is horizontal. θ = 90°, so W = F·d·cos 90° = 0 J. No work is done by gravity when there is no vertical displacement.
Kinetic energy depends on the square of speed — doubling speed quadruples Ek. This non-linear relationship has major consequences: a car at 100 km/h has four times the kinetic energy of one at 50 km/h, which is why stopping distances increase dramatically at higher speeds. The work-energy theorem connects work and kinetic energy directly.
When you accelerate an object, the distance it travels while accelerating also increases with speed. Work = force × distance — and since distance grows with v while force is constant, the total work (and thus energy stored) grows as v². This is a fundamental consequence of kinematics, not an arbitrary formula.
Negative work means the brakes removed energy from the car, converting it to heat. The brakes did −201.6 kJ of work.
b) An 80 J net work is done on a 2 kg object that starts from rest. What is its final speed?
c) Why does doubling speed quadruple kinetic energy?
a) Ek = ½ × 5 × 6² = ½ × 5 × 36 = 90 J
b) W = ΔEk = ½mv² − 0 → 80 = ½ × 2 × v² → v² = 80 → v = √80 ≈ 8.94 m/s
c) Ek = ½mv². If v doubles to 2v, then Ek = ½m(2v)² = ½m·4v² = 4(½mv²). The squaring of v means doubling v multiplies Ek by 4.
Potential energy is stored energy that can be converted to kinetic energy. Gravitational PE depends on height above a reference — you choose the reference. Spring PE stores energy when a spring is deformed from its natural length.
Only differences in height matter, not absolute height. A book on a table has higher Ep relative to the floor, but lower Ep relative to the ceiling. Since we always use ΔEp = mgΔh in energy conservation, the reference cancels out. Pick the level that makes h = 0 at the most convenient point (usually the lowest point in the problem).
b) A spring with k = 250 N/m is compressed 0.2 m. What is the elastic PE?
c) If the reference level is the table top instead of the floor, how does the Ep of a book sitting on the table change?
a) Ep = mgh = 70 × 9.8 × 12 = 8232 J = 8.232 kJ
b) Ee = ½kx² = ½ × 250 × (0.2)² = 125 × 0.04 = 5 J
c) The book's Ep becomes zero. When the reference level is set at the table top, h = 0 for the book. Ep = mgh = mg × 0 = 0. Only differences in height between two positions matter in energy calculations, so the choice of reference doesn't affect those differences.
In a closed system with no friction, total mechanical energy (Ek + Ep) is constant. Energy flows between kinetic and potential forms, but the total never changes. When friction exists, mechanical energy decreases — converted to thermal energy. The total energy of the universe is still conserved; we just lose useful mechanical energy.
At the top: Ek = 0 (dropped from rest), Ep = mgh. At the bottom: Ep = 0 (reference), Ek = ½mv².
v = √58.8 ≈ 7.67 m/s
b) A pendulum with L = 1.2 m is released from 40°. What is its speed at the bottom?
c) Why does a real pendulum eventually stop even though energy is "conserved"?
a) mgh = ½mv² → v = √(2 × 9.8 × 5) = √98 ≈ 9.9 m/s
b) h = 1.2(1 − cos 40°) = 1.2(1 − 0.766) = 1.2 × 0.234 = 0.2808 m
v = √(2 × 9.8 × 0.2808) = √5.503 ≈ 2.35 m/s
c) A real pendulum experiences air resistance and friction at the pivot. These forces do negative work on the pendulum, converting mechanical energy to thermal energy. Energy is still conserved in the universe — it just leaves the mechanical system as heat. The total energy decreases with each swing until it reaches zero.
Power tells you how fast energy is transferred. Two workers can do the same work — the one who finishes faster has greater power. Efficiency measures how much input energy becomes useful output. No real machine is 100% efficient: friction, heat, and sound always claim some energy.
P = W/t = (F·d)/t = F·(d/t) = F·v. This form is useful when you know a constant force and the speed at which it acts — for example, an engine maintaining a constant thrust at a given speed.
b) A 65 kg person climbs a 6 m ladder in 5 s. What is their power in watts and in horsepower (1 hp = 746 W)?
c) A machine receives 500 J of input energy and delivers 375 J of useful output energy. What is its efficiency?
a) P = W/t = 6000 / 40 = 150 W
b) W = mgh = 65 × 9.8 × 6 = 3822 J
P = 3822 / 5 = 764.4 W
In horsepower: 764.4 / 746 ≈ 1.02 hp
c) eff = (375 / 500) × 100% = 75%