16.1 Hooke’s Law: Stress and Strain Revisited
- An oscillation is a back and forth motion of an object between two points of deformation.
- An oscillation may create a wave, which is a disturbance that propagates from where it was created.
- The simplest type of oscillations and waves are related to systems that can be described by Hooke’s law:
where is the restoring force, is the displacement from equilibrium or deformation, and is the force constant of the system.
- Elastic potential energy stored in the deformation of a system that can be described by Hooke’s law is given by
16.2 Period and Frequency in Oscillations
- Periodic motion is a repetitious oscillation.
- The time for one oscillation is the period .
- The number of oscillations per unit time is the frequency .
- These quantities are related by
16.3 Simple Harmonic Motion: A Special Periodic Motion
- Simple harmonic motion is oscillatory motion for a system that can be described only by Hooke’s law. Such a system is also called a simple harmonic oscillator.
- Maximum displacement is the amplitude . The period and frequency of a simple harmonic oscillator are given by
and , where is the mass of the system.
- Displacement in simple harmonic motion as a function of time is given by
- The velocity is given by , where .
- The acceleration is found to be
16.4 The Simple Pendulum
- A mass suspended by a wire of length is a simple pendulum and undergoes simple harmonic motion for amplitudes less than about
The period of a simple pendulum is
where is the length of the string and is the acceleration due to gravity.
16.5 Energy and the Simple Harmonic Oscillator
- Energy in the simple harmonic oscillator is shared between elastic potential energy and kinetic energy, with the total being constant:
- Maximum velocity depends on three factors: it is directly proportional to amplitude, it is greater for stiffer systems, and it is smaller for objects that have larger masses:
16.6 Uniform Circular Motion and Simple Harmonic Motion
A projection of uniform circular motion undergoes simple harmonic oscillation.
16.7 Damped Harmonic Motion
- Damped harmonic oscillators have non-conservative forces that dissipate their energy.
- Critical damping returns the system to equilibrium as fast as possible without overshooting.
- An underdamped system will oscillate through the equilibrium position.
- An overdamped system moves more slowly toward equilibrium than one that is critically damped.
16.8 Forced Oscillations and Resonance
- A system’s natural frequency is the frequency at which the system will oscillate if not affected by driving or damping forces.
- A periodic force driving a harmonic oscillator at its natural frequency produces resonance. The system is said to resonate.
- The less damping a system has, the higher the amplitude of the forced oscillations near resonance. The more damping a system has, the broader response it has to varying driving frequencies.
- A wave is a disturbance that moves from the point of creation with a wave velocity .
- A wave has a wavelength , which is the distance between adjacent identical parts of the wave.
- Wave velocity and wavelength are related to the wave’s frequency and period by or
- A transverse wave has a disturbance perpendicular to its direction of propagation, whereas a longitudinal wave has a disturbance parallel to its direction of propagation.
16.10 Superposition and Interference
- Superposition is the combination of two waves at the same location.
- Constructive interference occurs when two identical waves are superimposed in phase.
- Destructive interference occurs when two identical waves are superimposed exactly out of phase.
- A standing wave is one in which two waves superimpose to produce a wave that varies in amplitude but does not propagate.
- Nodes are points of no motion in standing waves.
- An antinode is the location of maximum amplitude of a standing wave.
- Waves on a string are resonant standing waves with a fundamental frequency and can occur at higher multiples of the fundamental, called overtones or harmonics.
- Beats occur when waves of similar frequencies and are superimposed. The resulting amplitude oscillates with a beat frequency given by
16.11 Energy in Waves: Intensity
Intensity is defined to be the power per unit area:
and has units of .