A drawn bowstring holds energy: let it go, and the arrow flies. Model Xenia's practice bow as a spring with k = 100 N/m, drawn 0.20 m.
Last lesson, the truck spring's stored energy came off a graph: the area under its line. That works, but it is slow. Is there a formula instead?
Start from the numbers. At x = 0.20 m, the holding force is F = kx: 100 × 0.20 = 20, so 20 N.
The stored energy is the triangle's area under the force-extension line: average force × extension.
Average force: ½ × 20 = 10, so the energy is 10 × 0.20 = 2.0, which is 2.0 J.
Now do the same with symbols. The area is ½ × final force × extension, and the final force is kx.
So the stored energy is ½ × kx × x, which is ½kx².
This stored energy is called elastic potential energy, EPE for short.
Check it on the bow. x² is 0.20 × 0.20 = 0.04, and EPE = (½ × 100) × 0.04 = 2.0: the same 2.0 J the triangle gave.
Which relation gives the elastic potential energy stored in a spring at extension x?
A junior archery bow is modeled as a spring with k = 180 N/m, drawn 0.20 m. Take the draw as the extension. Calculate the EPE stored.
Write down the values:
k = 180 N/m, x = 0.20 m, EPE = ?
Write down the equation:
EPE = ½kx²
x² first: 0.20 × 0.20 = 0.04.
Substitute in the values, and calculate:
EPE = (½ × 180) × 0.04 = 3.6
The drawn bow stores 3.6 J.
A hatchback's trunk spring has k = 240 N/m and sits stretched 0.25 m while the trunk is open. Enter the EPE stored, in J.
A door-closer spring with k = 150 N/m is compressed 0.20 m as the door swings wide. Calculate the EPE stored.
Write down the values:
k = 150 N/m, x = 0.20 m, EPE = ?
Write down the equation:
EPE = ½kx²
x² first: 0.20 × 0.20 = 0.04.
Substitute in the values, and calculate:
EPE = (½ × 150) × 0.04 = 3.0
The compressed spring stores 3.0 J.
In EPE = ½kx², where does the ½ come from?
A climbing-wall safety spring has k = 400 N/m and is compressed 0.15 m when a climber steps on the pad. Enter the EPE stored, in J.
Which statement about a stretched spring's elastic potential energy is right?
A theater's stage-curtain spring, k = 600 N/m, sits stretched 0.40 m while the curtain is tied open. Enter the EPE stored, in J.