WAVES SIMPLE HARMONIC MOTION VIBRATIONS
(OSCILLATIONS) ·
Anything that moves back and forth, to or fro, side to
side, inoutin, or up or down is said to be vibrating or oscillating. ·
Time variations that repeat themselves at regular intervals: periodic or cyclic behavior. Fig. 1. Identify all the vibrations in the
pictures. A vibration is a periodic “wiggle” in time. A wave is a periodic “wiggle” in both space and time. The source of all waves is something that is
vibrating. Light and sound are
both vibrations that propagate through space as a wave, but are two very
different types of waves. 
VIBRATIONS (OSCILLATIONS) When an object is disturbed from its
equilibrium position, a restoring force acts on it to restore it back to the
equilibrium position. If the object over shoots the equilibrium position and
oscillates to and fro, the object is said to be
vibrating. For example, if you suspend a stone from a piece of string (simple
pendulum), the stone will vibrate back and forth when disturbed. When the
stone is attached to the end of a spring and held vertically, the stone will
bounce up and down when disturbed. Pendulums swing to and fro
with such regularity, they were used for a long time to control the motion of
most clocks. SIMPLE HARMONIC MOTION (SHM) To model vibrations, we need to setup
a simple model using approximations and simplifications. The simplest model
to describe vibrations is called simple harmonic motion. In this model, the
object will move backward and forward in a straight line about an equilibrium
position with a period which is independent of the magnitude of the
disturbance and the displacement of the object from its equilibrium position
can be described by a sinusoidal function. For the vertical
oscillations of an object, the frame of reference has the +Y axis pointing
upwards and the equilibrium position corresponds to the Origin. The position
of the object at any time t is given by the displacement . Displacement Velocity Acceleration Displacement amplitude is the maximum extent of a vibration or
oscillation, measured from the position of equilibrium. Velocity amplitude is the maximum speed of the oscillating
object. Acceleration amplitude is the maximum acceleration of the
oscillating object. ·
The amplitude is always a
positive number. The symbol is
often used for the amplitude. ·
For SHM the acceleration
is proportional to the displacement and it direction is opposite to the
displacement. Period is the time for one cycle of motion [s]. Frequency
is the number of cycles in one second [hertz
Hz 1 Hz = 1 s^{1}] 1
kHz = 10^{3} Hz (kilo) 1 MHz = 10^{6} Hz (mega) 1GHz = 10^{9} Hz (giga) Angular frequency [ rad.s^{1}]
Phase angle [ rad ] angle must be measured in radians and time in seconds The displacement , velocity and acceleration are all sinusoidal functions of time. 

A periodic signal is the recording of an
ECG.
The
brightness of stars varies periodically.

Fig. 2.
An animation for an object executing SHM in the vertical direction. 
Exercise 1 Watch the animation and check that the
graphs do successfully describe the motion of the object. Use the graphs to calculate and check
your answers for the estimation of the following parameters: ·
Period,
frequency and angular frequency ·
Amplitudes:
displacement, velocity and acceleration The slope of the tangent to a
displacement vs time graph gives the velocity. Verify from the graphs. The slope of the tangent to a velocity
vs time graph gives the acceleration. Verify. Predict the changes in the three
graphs if (1) The amplitude was decreased. (2) The period was decreased. 
Exercise 2
Use the graphs to calculate and check
your answers for the estimation of the following parameters: ·
Period,
frequency and angular frequency ·
Amplitudes:
displacement, velocity and acceleration 
Animation
produced with osc_shm_01A.m SHM
graphs produced with osc_shm_01.m If you have any feedback, comments, suggestions or corrections please email: Ian Cooper
School of Physics University
of Sydney ian.cooper@sydney.edu.au 