Energy equivalence in a perfect transfer
- When a force acts through a distance, the energy that leaves one store and enters another equals the work done by that force (mechanical pathway)
- In a perfect transfer (one with no friction or air resistance), no energy is dissipated as waste, so the full amount can be equated:
energy in one store before = energy in another store after
- The phrase "ignore air resistance" or "ignore frictional effects" in an exam question is the signal that you should use this energy-equivalence shortcut
Pendulum: GPE ↔ KE swap
- A swinging pendulum bob exchanges energy back and forth between its two stores on every swing:
- At the highest point of the swing, the bob is momentarily at rest. All its energy sits in the gravitational potential store
- At the lowest point of the swing, the bob is at its fastest. All its energy sits in the kinetic store
- For a frictionless pendulum, the maximum GPE at the top equals the maximum KE at the bottom:
m × g × h_max = ½ × m × v_max²
- The mass cancels from both sides, leaving:
v_max = √(2 × g × h_max)
Rollercoaster: same idea on a track
- A rollercoaster carriage starting at rest at the top of a tall hill arrives at the bottom with maximum speed; all the GPE lost during the descent has been converted to KE (ignoring friction and air drag)
- The maximum speed depends only on the drop in height, not on the route the track takes between the top and bottom
Example — a 0.20 kg tennis ball is dropped from rest at a height of 1.8 m above the floor. Air resistance is small enough to ignore. Take g = 10 N/kg. Calculate the speed of the ball just before it hits the floor.
- Set GPE lost = KE gained: m × g × h = ½ × m × v²
- The mass cancels: v² = 2 × g × h = 2 × 10 × 1.8 = 36
- v = √36 = 6.0 m/s