Learning Objectives
By the end of this section, you will be able to do the following:
- Explain Newton’s third law of motion with respect to stress and deformation
- Describe the restoring force and displacement
- Use Hooke’s law of deformation, and calculate stored energy in a spring
The information presented in this section supports the following AP® learning objectives and science practices:
- 3.B.3.3 The student can analyze data to identify qualitative or quantitative relationships between given values and variables (i.e., force, displacement, acceleration, velocity, period of motion, frequency, spring constant, string length, mass) associated with objects in oscillatory motion to use that data to determine the value of an unknown. (S.P. 2.2, 5.1)
- 3.B.3.4 The student is able to construct a qualitative and/or a quantitative explanation of oscillatory behavior given evidence of a restoring force. (S.P. 2.2, 6.2)
Newton’s first law implies that an object oscillating back and forth is experiencing forces. Without force, the object would move in a straight line at a constant speed rather than oscillate. Consider, for example, plucking a plastic ruler to the left as shown in Figure 16.2. The deformation of the ruler creates a force in the opposite direction, known as a restoring force. Once released, the restoring force causes the ruler to move back toward its stable equilibrium position, where the net force on it is zero. However, by the time the ruler gets there, it gains momentum and continues to move to the right, producing the opposite deformation. It is then forced to the left, back through equilibrium, and the process is repeated until dissipative forces dampen the motion. These forces remove mechanical energy from the system, gradually reducing the motion until the ruler comes to rest.
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 given to this relationship between force and displacement was Hooke’s law
Here, is the restoring force, is the displacement from equilibrium or deformation, and 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 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 are newtons per meter (N/m). For example, is directly related to Young’s modulus when we stretch a string. Figure 16.4 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 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 16.1 How Stiff Are Car Springs?
What is the force constant for the suspension system of a car that settles 1.20 cm when an 80-kg person gets in?
Strategy
Consider the car to be in its equilibrium position before the person gets in. The car then settles down 1.20 cm, which means it is displaced to a position . At that point, the springs supply a restoring force equal to the person’s weight We take this force to be in Hooke’s law. Knowing and , we can then solve the force constant .
Solution
- Solve Hooke’s law, , for .
16.2
Substitute known values and solve .
16.3
Discussion
Note that and 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.