Learning Objectives

Learning Objectives

Learning Objectives
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Concepts & Skills to Review

Hooke's law (Section 6.6)
graphical relationship of position, velocity, and acceleration (Sections 3.2 and 3.3)
elastic potential energy (Section 6.7)

Mastering the Concepts

A deformation is a change in the size or shape of an object.
When deforming forces are removed, an elastic object returns to its original shape and size.
Hooke's law, in a generalized form, says that the deformation of a material (measured by the strain) is proportional to the magnitude of the forces causing the deformation (measured by the stress). The definitions of stress and strain are as given in the following table.
If the tensile or compressive stress exceeds the proportional limit, the strain is no longer proportional to the stress. The solid still returns to its original length when the stress is removed as long as the stress does not exceed the elastic limit. If the stress exceeds the elastic limit, the material is permanently deformed. For larger stresses yet, the solid fractures when the stress reaches the breaking point. The maximum stress that can be withstood without breaking is called the ultimate strength.
Vibrations occur in the vicinity of a point of stable equilibrium. An equilibrium point is stable if the net force on an object when it is displaced from equilibrium points back toward the equilibrium point. Such a force is called a restoring force since it tends to restore equilibrium.
Simple harmonic motion is periodic motion that occurs whenever the restoring force is proportional to the displacement from equilibrium. In SHM, the position, velocity, and acceleration as functions of time are sinusoidal (i.e., sine or cosine functions). Any oscillatory motion is approximately SHM if the amplitude is small, because for small oscillations the restoring force is approximately linear.
The period T is the time taken by one complete cycle of oscillation. The frequency f is the number of cycles per unit time:

The angular frequency is measured in radians per unit time:

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The maximum velocity and acceleration in SHM are

where w is the angular frequency. The acceleration is proportional to and in the opposite direction from the displacement:
The equations of motion for SHM are

In either case, the velocity is one-quarter cycle ahead of the position and the acceleration is one-quarter cycle ahead of the velocity
The period of oscillation for a mass-spring system is

For a simple pendulum it is

and for a physical pendulum it is

In the absence of dissipative forces, the total mechanical energy of a simple harmonic oscillator is constant and proportional to the square of the amplitude:

where the potential energy has been chosen to be zero at the equilibrium point. At any point, the sum of the kinetic and potential energies is constant:

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	Course-wide Content
		MCAT Practice Test MCAT Review Inside Cover Content f...

	Quizzes
		Multiple Choice Quiz
	More Resources
		Learning Objectives Selected Solutions More Resources
	Animated Examples
		Cornell Interactives