16.1 Hooke’s law: stress and strain revisited

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 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 [link] . 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 was given to this relationship between force and displacement was Hooke’s law:

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.