Hooke’s Law: Stress and Strain Revisited
Lumen Learning
Learning Objectives
By the end of this section, you will be able to:
- Explain Newton’s third law of motion with respect to stress and deformation.
- Describe the restoration of force and displacement.
- Calculate the energy in Hook’s Law of deformation, and the stored energy in a string.
The simplest oscillations occur when the restoring force is directly proportional to displacement. When stress and strain were covered in Newton’s Third Law of Motion, the name was given to this relationship between force and displacement was Hooke’s law: F = −kx.
Here, F is the restoring force, x is the displacement from equilibrium or deformation, and k is a constant related to the difficulty in deforming the system. The minus sign indicates the restoring force is in the direction opposite to the displacement.
The force constant k is related to the rigidity (or stiffness) of a system—the larger the force constant, the greater the restoring force, and the stiffer the system. The units of k are newtons per meter (N/m). For example, k is directly related to Young’s modulus when we stretch a string. Figure 3 shows a graph of the absolute value of the restoring force versus the displacement for a system that can be described by Hooke’s law—a simple spring in this case. The slope of the graph equals the force constant k in newtons per meter. A common physics laboratory exercise is to measure restoring forces created by springs, determine if they follow Hooke’s law, and calculate their force constants if they do.
Example 1. How Stiff Are Car Springs?
Strategy
Consider the car to be in its equilibrium position x=0 before the person gets in. The car then settles down 1.20 cm, which means it is displaced to a position x = −1.20 × 10−2 m. At that point, the springs supply a restoring force F equal to the person’s weight w = mg = (80.0 kg)(9.80 m/s2) = 784 N. We take this force to be F in Hooke’s law. Knowing F and x, we can then solve the force constant k.
Solution
Solve Hooke’s law, F = −kx, for k:
Substitute known values and solve k:
Discussion
Note that F and x have opposite signs because they are in opposite directions—the restoring force is up, and the displacement is down. Also, note that the car would oscillate up and down when the person got in if it were not for damping (due to frictional forces) provided by shock absorbers. Bouncing cars are a sure sign of bad shock absorbers.
Energy in Hooke’s Law of Deformation
In order to produce a deformation, work must be done. That is, a force must be exerted through a distance, whether you pluck a guitar string or compress a car spring. If the only result is deformation, and no work goes into thermal, sound, or kinetic energy, then all the work is initially stored in the deformed object as some form of potential energy. The potential energy stored in a spring is . Here, we generalize the idea to elastic potential energy for a deformation of any system that can be de