Wells is packing his truck. He stretches the cargo-net spring out to 0.40 m of extension to hook it over the load.
At the start, holding it takes no force at all; by the end, it takes 80 N.
Stretching the spring took work: a force acted through a distance. But the force kept changing, from 0 N up to 80 N. How much work did the whole stretch take?
The truck spring's numbers: the force climbed steadily from 0 N at no extension to 80 N at 0.40 m.
Here is the force-extension graph of the stretch: a straight line from the origin up to 80 N at 0.40 m.
The force grew steadily, so its average over the stretch is half the final value: ½ × 80 = 40, an average of 40 N.
Work done = average force × distance moved: 40 × 0.40 = 16, so the stretch took 16 J.
Now look at the graph. The region under the line is a triangle: base 0.40 m, height 80 N.
A triangle's area is half of base × height: the same multiplication again. The area is exactly the work done.
A second spring: its line reaches 20 N at 0.10 m of extension.
Average force ½ × 20 = 10, so the stored energy is 10 × 0.10 = 1.0, which is 1.0 J: the triangle's area.
The force-extension graph shown is a straight line through the origin reaching 40 N at 0.20 m of extension. Enter the energy stored at 0.20 m, in J.
A spring's force-extension line rises straight from the origin. The stored energy is half of final force × extension, not the full product. Why the half?
A force-extension graph for a bike-rack spring is a straight line through the origin reaching 60 N at 0.30 m of extension. Calculate the energy stored at 0.30 m.
Write down the values, read from the line:
final force = 60 N, extension = 0.30 m
The stored energy is the area of the triangle under the line:
stored energy = ½ × final force × extension
Substitute in the values, and calculate:
stored energy = (½ × 60) × 0.30 = 9.0
The spring stores 9.0 J.
A dog-leash spring's graph is a straight line through the origin reaching 24 N at 0.050 m of extension. Calculate the energy stored at 0.050 m.
Write down the values, read from the line:
final force = 24 N, extension = 0.050 m
The stored energy is the triangle's area:
stored energy = ½ × final force × extension
Substitute in the values, and calculate:
stored energy = (½ × 24) × 0.050 = 0.60
The spring stores 0.60 J.
The force-extension graph shown is a straight line through the origin reaching 28 N at 0.50 m of extension. Enter the energy stored at 0.50 m, in J.
The force-extension graph shown is a straight line through the origin reaching 50 N at 0.24 m of extension. Enter the energy stored at 0.24 m, in J.
The graph shown carries two springs' lines: a cart spring reaching 40 N at 0.30 m of extension, and a gate spring reaching 40 N at 0.60 m. Which spring stores more energy at the extension shown?
A robotics team logs a gripper-spring test. The force-extension graph shown is a straight line through the origin reaching 18 N at 0.20 m of extension. Enter the energy stored at 0.20 m, in J.