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## Physics library

### Course: Physics library>Unit 5

Lesson 2: Springs and Hooke's law

# What is elastic potential energy?

Learn what elastic potential energy means and how to calculate it.

# What is elastic potential energy?

Elastic potential energy is energy stored as a result of applying a force to deform an elastic object. The energy is stored until the force is removed and the object springs back to its original shape, doing work in the process. The deformation could involve compressing, stretching or twisting the object. Many objects are designed specifically to store elastic potential energy, for example:
• The coil spring of a wind-up clock
• An archer's stretched bow
• A bent diving board, just before a divers jump
• The twisted rubber band which powers a toy airplane
• A bouncy ball, compressed at the moment it bounces off a brick wall.
An object designed to store elastic potential energy will typically have a high elastic limit, however all elastic objects have a limit to the load they can sustain. When deformed beyond the elastic limit, the object will no longer return to its original shape. In earlier generations, wind-up mechanical watches powered by coil springs were popular accessories. Nowadays, we don't tend to use wind-up smartphones because no materials exist with high enough elastic limit to store elastic potential energy with high enough energy density.

# How can we calculate elastic potential energy for an ideal spring?

Our article on Hooke's law and elasticity discusses how the magnitude of the force F due to an ideal spring depends linearly on the length it has been compressed or expanded delta, x,
F, equals, k, dot, delta, x
where k is some positive number known as the spring constant. The spring force is a conservative force and conservative forces have potential energies associated with them.
From the definition of work we know that the area under a force vs displacement graph gives the work done by the force. Figure 1 shows a plot of force vs displacement for a spring. Because the area under the curve is a triangle and no energy is lost in an ideal spring, the elastic potential energy U can be found from the work done
\begin{aligned}U &= \frac{1}{2} (\Delta x) \cdot k (\Delta x) \\ &= \boxed{\frac{1}{2} k (\Delta x)^2} \end{aligned}
Figure 1: The work done by a force on an ideal spring.
Figure 1: The work done by a force on an ideal spring.
Exercise 1: A truck spring has a spring constant of 5, dot, 10, start superscript, 4, end superscript, space, N, slash, m. When unloaded, the truck sits 0.8 m above the road. When loaded with goods, it lowers to 0.7 m above the ground. How much potential energy is stored in the four springs?
Exercise 2a: A trained archer has the ability to draw a longbow with a force of up to 300 N, extending the string back by 0.6 m. Assuming the bow behaves like an ideal spring, what spring constant would allow the archer to make use of his full strength?
A drawn bow.
Figure 2: A drawn bow, as used in exercise 2.
Exercise 2b: What potential energy is stored in the bow when it is drawn?
Exercise 2c: Assuming the arrow has a mass of 30 g, approximately what speed will it be fired at?
Exercise 2d: Suppose that measurements from a high speed camera show the arrow to be moving at a somewhat slower speed than predicted by conservation of energy. Is there any work being done that we have not accounted for?

# What about real elastic materials?

In our article on Hooke's law and elasticity we discuss how real springs only obey Hooke's law over some particular range of applied force. Some elastic materials such as rubber bands and flexible plastics can function as springs but often have hysteresis; this means the force vs extension curve follows a different path when the material is being deformed compared to when it is relaxing back to its equilibrium position.
Fortunately, the basic technique of applying the definition of work that we employed for an ideal spring also works for elastic materials in general. The elastic potential energy can always be found from the area under the force vs extension curve, regardless of the shape of the curve.
In our earlier analysis, we have considered the ideal spring as a one-dimensional object. In reality, elastic materials are three dimensional. It turns out that the same procedure still applies. The equivalent to the force vs extension curve is the stress vs strain curve.
Where a three-dimensional elastic material obeys Hooke's law,
E, n, e, r, g, y, slash, v, o, l, u, m, e, equals, start fraction, 1, divided by, 2, end fraction, left parenthesis, S, t, r, e, s, s, dot, S, t, r, a, i, n, right parenthesis
Exercise 3: Figure 3 shows a stress vs strain plot for a rubber band. As it is stretched (loaded), the curve takes the upper path. Because the rubber band is not ideal, it delivers less force for a given extension when relaxing back (unloaded). The purple shaded area represents the elastic potential energy at maximum extension. The difference in area between the loaded and unloaded case is shown in yellow. This represents the energy which is lost to heat as the band is cycled between stretched and relaxed.
If the rubber band has length 100, space, m, m, width 10, space, m, m and thickness 1, space, m, m how much heat is generated in the band as it is stretched and released?
Figure 3: Force vs extension curve for a rubber band. Vertical and horizontal gridlines at 0.05 units.
Figure 3: Force vs extension curve for a rubber band. Vertical and horizontal gridlines at 0.05 units.