Vertical spring — the four energies
● VERBATIM — examined word-for-wordTrack GPE, EPE, KE, Total (columns of the classic table). Lowest point: EPE max, GPE min (reference 0), KE 0. Equilibrium: KE max. Highest point: GPE max, EPE min (not zero if spring still stretched), KE 0. Total constant. Shortcut for everything else: measure x from equilibrium and use the combined PE = ½kx² — gravity is already absorbed into the equilibrium position, so all standard SHM formulas apply unchanged.
What this actually means
Four ledgers run at once: gravitational PE, elastic PE, kinetic energy, total. Lowest point: EPE max, GPE min, KE zero. Equilibrium: KE max. Highest point: GPE max, KE zero, and EPE at its minimum but NOT necessarily zero, because the spring can still be stretched there.
The shortcut that dissolves the whole table: measure x from the equilibrium position and use one combined PE = ½kx². Gravity is already absorbed into where equilibrium sits.
With that shortcut, every standard SHM formula applies unchanged, and the GPE-versus-EPE bookkeeping the question was designed to drown you in never starts.
Chasing GPE and EPE separately — measure x from equilibrium and use the combined ½kx².
Prove it — watch it be true
- Watch the energy bars through one cycle: KE maxes at the centre, PE at both extremes
- Note the PE bar is one combined quantity, measured from equilibrium
- That single bar is the ½kx² shortcut working