The motion of an object under the influence of either

a force characterized by Hook's law:

*F = -kx*

or *equivalently*

a potential given by

*PE*_{sho}* = (1/2)kx*^{2}

where *sho* stands for simple harmonic oscillator.

Force and potential (energy) are two ways to describe a
conservative interaction.

Force is equal to the negative of the slope of the corresponding
potential.

Examples to be studied: The motion of an object attached to a
spring and the motion of a pendulum.

*Almost all types of motion that have an equilibrium
position can be approximated by an simple harmonic oscillator if
their amplitudes are sufficiently small.*

The understanding of simple harmonic motion is also the basis for
understanding the wave motion.

Understanding based on Newton's second law:

*a = F/m = -(k/m)x*

- The motion is back and forth around the equilibrium position.
- Acceleration is zero at the equilibrium position (
*where the velocity is greatest*), and it is greatest at the maxima distances away from the equilibrium (*where the velocity is zero*).

Understanding based on the conservation of energy:

a) No friction (no damping)

*E = KE + PE = const*

*(1/2)mv*^{2}* + (1/2)kx*^{2}*
= E = const*

from which it is clear that energy can be written in terms of the
amplitude as

*E = (1/2)kA*^{2}

It is also clear that the maximum velocity is achieved at the
equilibrium position (x=0) and

*E = (1/2)mv*_{max}^{2}

*(1/2)mv*_{max}^{2}* = E =
(1/2)kA*^{2}

In a simple harmonic motion, kinetic energy and potential energy
are converting back and forth into each other. The energy is
purely kinetic at the equilibrium position, and it is pure
potential energy at the maxima positions.

b) With friction

*D**E = W*_{friction}*
= -fs*

Example 13.4 Motion with and without friction

Amplitude *A* and frequency *f*.

Amplitude: the maximum distance traveled by an object away from
its equilibrium position.

Amplitude depends on the initial condition, i.e. how you get the
motion started; how much energy you put into the system.

Frequency: the number of cycles per unit time.

Frequency is an intrinsic property of the oscillator and is
independent of the initial conditions (as long as you do not *stretch*
it too hard).

*f = (1/2**p**)(k/m)*^{1/2}

This equation is derived by solving the Newton's equation of
motion. The solution gives us position as a function of time:

*x = A cos **w **t*

where

*w** = 2**p**f
= (k/m)*^{1/2}

is called the angular frequency (rad/s).

There is another parameter that is related to frequency. It is
the period T.

Period *T*: time it takes to complete one cycle.

*T = 1/f = 2**p**(m/k)*^{1/2}

(f = 1/T may be even easier to see).

Example 13.7 The vibrating mass-spring system

***

There is in fact a third parameter that characterizes a simple
harmonic motion. It is called *phase *and is determined by
initial conditions. The concept of phase is very important in
understanding the interference between two waves. Briefly, the
more general description of displacement as a function of time is

*x = A cos(**w **t*+*f** )*

in which f is called the phase.

***

A pendulum is a good example of the motion near an equilibrium
being well approximated by a simple harmonic oscillator. For
small angles, the restoring force for a pendulum is

*F = -mg sin **q » **-mg **q **= -(mg/L)s*

which corresponds to a simple harmonic oscillator with an
effective spring constant

*k = mg/L*

The period of a pendulum is therefore given by

*T = 2**p**(L/g)*^{1/2}

Example 13.8 What is the height of that tower?

Propagation of energy or disturbance.

A spatially correlated oscillations.

Amplitude and two out of the following three parameters:

- frequency
*f* - velocity
*v* - wavelength
*l*

which are related by

*v = f **l*

Using light (an electromagnetic wave) as an example, the
amplitude is the parameter that determines the intensity of the
light and frequency (or wavelength) is the parameter that
determines the color of the light.

Example 13.9 A traveling wave

Transverse wave: a wave in which the displacement is
perpendicular to the direction of propagation.

Longitudinal wave: a wave in which the displacement is parallel
to the direction of propagation.

Figure 13.19 is a illustration of two types of waves.

Examples

a) Light (electromagnetic) wave

Light waves of all frequency travel at the same speed in vacuum.
The speed of light in vacuum:

c = 2.99792458 ´ 10^{8}
m/s

The speed of light in a medium is generally frequency-dependent.
This phenomenon is called (or gives rise to what is called)
dispersion.

Light wave is transverse in nature.

b) Transverse wave on a string

The speed of a transverse wave on a string is determined by

*v = (F/**m**)*^{1/2}

in which F is the tension in the string and m
is mass per unit length.

Example 13.12 A pulse traveling on a string

c) Sound wave

A sound wave in air is a longitudinal wave.

Audible sound has a frequency range of 20 Hz to 20,000 Hz.

Doppler effect

The displacement at a point which is under the influence of
multiple waves is equal to the sum of displacements induced by
each individual wave at that point.

Constructive interference: The influences of two waves add to
each other.

Destructive interference: The influences of two waves cancel each
other.

Figure 13.25 illustrates constructive interference and Figure
13.26 illustrates destructive interference.

The two waves in Figure 13.25 are said to be in phase
(oscillations induced by both waves at a specific point reach
their maxima at the same time).

The two waves in Figure 13.26 are said to be 180 degrees out of
phase (oscillation induced by one of waves is at its minimum when
the oscillation induced by the other is at its maximum).

***

Waves are described mathematically by so called wave equations.
Waves satisfying the superposition principle are described by
equations which are *linear*. Maxwell's equation for an
electromagnetic wave in vacuum and the Schrodinger equation in
quantum physics are examples of wave equations, both of which are
linear.

A medium may become nonlinear if the wave traveling through it is
too intense (amplitude too big).

***

Contents | Thermodynamics | Vibrations and
Waves | E&M

Copyright © 1997 by Bo Gao.
All rights reserved.