Energy is defined as the capacity to do work. It's a scalar quantity that exists in many forms: kinetic, potential (gravitational, elastic, electric), thermal, chemical, nuclear, electromagnetic, and others.
The fundamental principle is that energy is conserved: it can transform from one form to another and transfer between objects, but the total energy of an isolated system remains constant. Energy cannot be created or destroyed.
SI unit: joule (J), where 1 J = 1 N·m = 1 kg·m²/s²
This conservation principle provides powerful problem-solving techniques, often simpler than tracking forces through every moment of motion.
W = F·d = Fd cos θ
F = force magnitude, d = displacement magnitude, θ = angle between force and displacement
Definition: Work is done when a force acts on an object, and the object moves through a displacement. Only the component of force parallel to the displacement does work.
Sign conventions:
Example 1: You push a box with 50 N at 30° below horizontal, moving it 4 m horizontally. Work done?
W = Fd cos θ = (50)(4)(cos 30°) = (50)(4)(0.866) = 173.2 J
Example 2: You carry a 20 kg backpack (weight = 196 N) horizontally for 100 m at constant velocity. Work done by upward supporting force?
Force is vertical, displacement is horizontal → θ = 90° → W = 0 J
Work Done Against Gravity: When lifting vertically at constant velocity:
W = Fd = mgh
This work equals the increase in gravitational potential energy.
KE = ½mv²
m = mass (kg), v = speed (m/s)
Kinetic energy is the energy possessed by an object due to its motion. It is a scalar quantity (always positive for moving objects).
Kinetic energy is proportional to mass but proportional to the square of speed. Doubling speed quadruples kinetic energy.
Example: A 1200 kg car travels at 25 m/s. Calculate its kinetic energy.
KE = ½(1200)(25)² = ½(1200)(625) = 375,000 J = 375 kJ
To stop this car, the brakes must dissipate 375 kJ of energy, converting it to thermal energy.
W_net = ΔKE = KE_f - KE_i = ½mv_f² - ½mv_i²
Statement: The net work done by all forces acting on an object equals the change in its kinetic energy.
This theorem connects the concepts of force (through work) and motion (through kinetic energy). It's often more efficient than using F = ma with kinematic equations.
Example: A 0.5 kg hockey puck slides with initial speed 8 m/s. Kinetic friction does -12 J of work. Find final speed.
KE_i = ½(0.5)(8)² = 16 J
KE_f = KE_i + W = 16 + (-12) = 4 J
½(0.5)v_f² = 4 → v_f² = 16 → v_f = 4 m/s
GPE = mgh
m = mass (kg), g = 9.81 m/s², h = height above reference level (m)
Gravitational potential energy is the energy possessed by an object due to its position in a gravitational field.
The reference level is arbitrary. Only changes in gravitational potential energy have physical significance.
ΔGPE = mg(h_f - h_i) = mgΔh
Example: A 2 kg book on a shelf 1.5 m above the floor. Taking floor as reference, find GPE.
GPE = mgh = (2)(9.81)(1.5) = 29.43 J
If it falls, this energy converts to kinetic energy.
EPE = ½kx²
k = spring constant (N/m), x = extension/compression from natural length (m)
Elastic potential energy is the energy stored in an elastic object (spring, rubber band) when deformed from its natural length.
Hooke's Law: F = -kx (restoring force is proportional to displacement)
Example: A spring with k = 200 N/m is compressed by 0.1 m. Calculate stored elastic potential energy.
EPE = ½(200)(0.1)² = ½(200)(0.01) = 1 J
Energy cannot be created or destroyed; it can only be transformed from one form to another or transferred between objects. The total energy of an isolated system remains constant.
E_i = E_f
KE_i + GPE_i + EPE_i = KE_f + GPE_f + EPE_f
Example: A 0.5 kg ball drops from rest at height 2 m. Find speed just before hitting ground. (Ignore air resistance)
Initial: KE_i = 0, GPE_i = (0.5)(9.81)(2) = 9.81 J
Final: KE_f = ½(0.5)v², GPE_f = 0
9.81 = ½(0.5)v² → v² = 39.24 → v = 6.26 m/s
Work done is independent of path; depends only on initial and final positions.
Examples: Gravity, elastic spring force, electrostatic force
Potential energy functions can be defined.
Work done depends on the path taken.
Examples: Friction, air resistance, tension
Convert mechanical energy to thermal energy.
W_non-conservative = ΔE_mechanical
E_i + W_non-conservative = E_f
Momentum conserved: Σp_before = Σp_after
Kinetic energy conserved: ΣKE_before = ΣKE_after
Momentum conserved: Σp_before = Σp_after
Kinetic energy NOT conserved: ΣKE_before > ΣKE_after
"Lost" KE → thermal energy, sound, deformation
Energy dissipated = KE_initial - KE_final
P = W/Δt = ΔE/Δt
For constant force and constant velocity: P = Fv cos θ
If force is in direction of motion: P = Fv
SI unit: watt (W), where 1 W = 1 J/s
Common units:
Example 1: An elevator motor lifts a 500 kg elevator 20 m vertically in 10 s. Minimum power required?
W = mgh = (500)(9.81)(20) = 98,100 J
P = W/t = 98,100/10 = 9,810 W ≈ 9.8 kW
Example 2: A car engine provides 40 kW while cruising at constant 30 m/s. Find forward force.
P = Fv → 40,000 = F(30) → F = 1333 N
η = (E_useful output / E_total input) × 100%
or η = (P_useful output / P_total input) × 100%
Efficiency is the ratio of useful energy output to total energy input, expressed as a percentage.
Efficiency is always less than 100% for real machines due to energy losses (primarily to thermal energy via friction).
Example: An electric motor draws 200 W but provides only 150 W of mechanical power. Find efficiency.
η = (150/200) × 100% = 75%
The remaining 25% (50 W) becomes thermal energy, warming the motor.
| Quantity | Symbol | Unit | Formula |
|---|---|---|---|
| Work | W | J (joule) | W = Fd cos θ |
| Kinetic Energy | KE | J | KE = ½mv² |
| Gravitational PE | GPE | J | GPE = mgh |
| Elastic PE | EPE | J | EPE = ½kx² |
| Power | P | W (watt) | P = W/t = Fv |
| Efficiency | η | % | η = (E_out/E_in) × 100% |