Conservation of Mechanical Energy
Senior 2 · Physics
Conservation of Mechanical Energy
Lesson 4 of 4
22 min
Much of Rwanda's electricity comes from falling water, at stations such as Nyabarongo I and the Rusumo Falls plant on the Akagera River. The physics behind them is the energy exchange you met in the last lesson: water high up has potential energy; as it falls it gains kinetic energy; a turbine and a generator turn that into electrical energy.
This lesson applies the conservation law to real machines and real journeys, and deals honestly with friction: the reason the real world never quite matches the ideal answer.
Water stored behind a dam flows down a steep pipe (a penstock), reaches the bottom moving fast, spins a turbine, and the turbine turns a generator. The main energy chain is:
gravitational PE → kinetic energy → electrical energy
Worked example. 1000 kg of water (1 m³) falls through a height of 50 m. Take g = 9.8 m/s².
PE lost = mgh = 1000 × 9.8 × 50 = 490 000 J (490 kJ) per cubic metre.
If 20 m³ of water pass through the turbines every second, the water delivers 20 × 490 000 = 9 800 000 J every second, that is 9.8 MW. A real station delivers less than this, because some energy is always turned into heat and sound in the pipes, turbine and generator.
In real life, friction and air resistance act on every moving object. They do not destroy energy. They turn some of the mechanical energy into thermal energy (heat) and sound. Total energy is still conserved, but mechanical energy is not:
ME at start = ME at finish + energy converted by friction
Worked example. Gatete and his bicycle have a total mass of 80 kg. He starts from rest at the top of a hill 20 m high and coasts down to Nyabugogo without pedalling. At the bottom his speed is 14 m/s.
Without friction he would have reached √(2 × 9.8 × 20) ≈ 19.8 m/s. Only half of the potential energy (7840 ÷ 15 680 = 50%) became kinetic energy. The rest warmed his brakes, tyres, bearings and the air.
Tip
Read the key words. "Smooth", "frictionless" or "neglect air resistance" means: use KE₁ + PE₁ = KE₂ + PE₂.
If the question gives a real final speed that is lower than your ideal answer, the difference in mechanical energy is the work done against friction. Calculate it as (PE lost) − (KE gained), and say where it went: heat and sound.
Take g = 9.8 m/s².
500 kg of water falls 30 m down a penstock. Ignoring friction, how much kinetic energy does it have at the bottom?
Solution: KE gained = PE lost = mgh = 500 × 9.8 × 30 = 147 000 J (147 kJ).
A 0.50 kg ball is dropped from 10 m and hits the ground at 12 m/s. How much energy was converted by air resistance?
Solution: PE lost = 0.50 × 9.8 × 10 = 49 J. KE gained = ½ × 0.50 × 12² = 36 J. Energy converted to heat and sound = 49 − 36 = 13 J.
A frictionless roller-coaster car starts from rest at the top of a 30 m hill. The next hill is 35 m high. Can the car get over it? Explain.
Solution: No. The car's total mechanical energy equals its PE at 30 m. To reach 35 m it would need more PE than the total energy it has. It would stop before the top and roll back.
A pendulum released from 0.45 m above its lowest point rises only to 0.40 m on the other side. What percentage of its mechanical energy was lost in that swing?
Solution: PE is proportional to height, so the fraction lost = (0.45 − 0.40) ÷ 0.45 = 0.111, that is about 11%. It became heat in the air and at the pivot.
Give two reasons why the electrical energy from a hydropower station is less than the potential energy lost by the water.
Solution: Any two of: friction between the water and the penstock walls turns some energy into heat; friction in the turbine and generator bearings produces heat and sound; the generator's coils warm up; the water leaving the turbine still has some kinetic energy.
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