Lesson 3 · 14 min
Conservation and efficiency
Energy is conserved. Useful energy is not. That difference is the entire thermal sciences.
When you may write E_i = E_f
If only conservative forces do work, mechanical energy is constant: KE + PE is the same at two points. If non-conservative forces do work, W_nc = ΔME. That W_nc is usually negative — friction, plastic hinges, heat.
Efficiency η = useful output / required input. It is always less than one for a real cyclic machine. A transformer can beat 98%. A gasoline engine is lucky to see 30% at the crank.
KE_i + PE_i + W_nc = KE_f + PE_f
W_nc is work by non-conservative forces, typically negative.
Worked example
Roller into a stop
- Given: m = 1200 kg
- Given: v = 8.0 m/s
- Given: spring bumper k = 4.0×10⁴ N/m
- Given: ignore friction
Find: Maximum spring compression
- ½mv² = ½kx²
- x = v √(m/k) = 8 √(1200 / 4.0e4) = 8 √0.03 = 8 × 0.173
1.39 m (a long bumper — or a stiffer k)
Check
A machine lifts 200 kg through 3.0 m using 8.0 kJ of electrical input. Efficiency?