Sometimes the stored energy is the known number, and the spring constant or the extension is the unknown. A door closer must store 2.0 J when its spring is compressed 0.10 m. What spring constant does it need?
The numbers: EPE must be 2.0 J at x = 0.10 m. The unknown is k.
Start from the equation you know: EPE = ½kx².
Make k the subject: double both sides, then divide by x².
That gives k = 2 × EPE ÷ x².
For the door closer: double the energy first: 2 × 2.0 = 4.0. Now divide by x twice, once for each factor in x²: 4.0 ÷ 0.10 = 40, and 40 ÷ 0.10 = 400. So k = 400 N/m.
Make x the subject the same way: x² = 2 × EPE ÷ k. Then take the square root.
That gives x = √(2 × EPE ÷ k).
The forward direction works as before: k = 400 N/m at 0.10 m stores (½ × 400 × 0.10) × 0.10 = 2.0, so 2.0 J.
A bike kickstand spring with k = 100 N/m must store 4.5 J. Calculate the extension needed.
Write down the values:
EPE = 4.5 J, k = 100 N/m, x = ?
Write down the equation and make x the subject:
x = √(2 × EPE ÷ k)
Double the energy: 2 × 4.5 = 9.0.
Divide by k: 9.0 ÷ 100 = 0.09. That is x².
Find x: square root both sides.
x = √(0.09) = 0.30
The extension needed is 0.30 m.
An office-chair spring stores 6.4 J when compressed 0.40 m. Calculate its spring constant.
Write down the values:
EPE = 6.4 J, x = 0.40 m, k = ?
Write down the equation and make k the subject:
k = 2 × EPE ÷ x²
Double the energy: 2 × 6.4 = 12.8.
x² is 0.40 × 0.40 = 0.16.
Substitute in the values, and calculate:
k = 12.8 ÷ 0.16 = 80
The spring constant is 80 N/m.
The questions below mix the three directions. When x is the unknown, take the square root last.
A spring with k = 50 N/m must store 1.0 J. Which arrangement finds the extension?
Use your arrangement on the same spring: k = 50 N/m, EPE = 1.0 J. Enter the extension, in m.
A trunk strut's spring stores 2.7 J at 0.30 m of compression. Enter its spring constant, in N/m.
A screen-roller spring with k = 128 N/m must store 4.0 J. Enter the extension needed, in m.
A recliner's spring stores 9.0 J when stretched 0.30 m. Enter its spring constant, in N/m.
A warehouse door's spring, k = 1200 N/m, sits stretched 0.30 m while the door stands open. Enter the EPE stored, in J.