Sometimes the stored energy is known, and the height or the mass is the unknown. A zip-line platform must give a 3.0 kg test mass 150 J of GPE, measured from the ground. How high must the platform be?
Estimate first with plain numbers. The test mass's weight: 3.0 × 9.8 = 29.4 N.
Every meter of height stores 29.4 J with it. Banking 150 J takes a little over five meters, because 29.4 × 5.0 = 147.
To get the exact height, rearrange GPE = mgh.
Make h the subject: divide both sides by mg.
That gives h = GPE ÷ (mg).
Make m the subject: divide both sides by gh.
That gives m = GPE ÷ (gh).
For the platform: h = 150 ÷ 29.4 = 5.1, so the platform must be 5.1 m high.
Check it against the estimate: just over five meters. It fits.
A crane must give a 40 kg beam 7840 J of GPE, measured from the ground. Take g = 9.8 N/kg. Calculate the height the beam must reach.
Write down the values:
GPE = 7840 J, m = 40 kg, g = 9.8 N/kg, h = ?
Write down the equation and make h the subject:
h = GPE ÷ (mg)
The divisor mg: 40 × 9.8 = 392.
Substitute in the values, and calculate:
h = 7840 ÷ 392 = 20
The beam must reach 20 m.
A stage counterweight hangs 1.5 m above the stage floor and stores 58.8 J of GPE, measured from that floor. Take g = 9.8 N/kg. Calculate the counterweight's mass.
Write down the values:
GPE = 58.8 J, g = 9.8 N/kg, h = 1.5 m, m = ?
Write down the equation and make m the subject:
m = GPE ÷ (gh)
The divisor gh: 9.8 × 1.5 = 14.7.
Substitute in the values, and calculate:
m = 58.8 ÷ 14.7 = 4.0
The counterweight's mass is 4.0 kg.
The questions below mix the two rearrangements; take g = 9.8 N/kg throughout.
A 2.0 kg bag must store 88.2 J of GPE. Which arrangement finds the height needed?
Use your arrangement on the same bag: m = 2.0 kg, GPE = 88.2 J, g = 9.8 N/kg. Enter the height, in m.
A crate on a 2.0 m platform stores 137.2 J of GPE, measured from the ground. Take g = 9.8 N/kg. Enter the crate's mass, in kg.
A 2.5 kg hanging sign must store 220.5 J of GPE, measured from the sidewalk. Take g = 9.8 N/kg. Enter the height at which it must hang, in m.
A speaker box mounted 2.0 m up a gym wall stores 156.8 J of GPE, measured from the gym floor. Take g = 9.8 N/kg. Enter the speaker's mass, in kg.
Yael's climbing gym keeps a 12 kg crash pad on a storage ledge. The pad stores 705.6 J of GPE, measured from the gym floor. Take g = 9.8 N/kg. Enter the ledge's height, in m.