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PHYSICS 215
SIMPLE HARMONIC MOTION
Object:
The object of this experiment is to predict and measure properties of systems moving in
simple harmonic motion: a simple pendulum; a physical pendulum; and a linear mass and
springs oscillator.
Apparatus:
Rotary Motion Sensor
Rod and Masses
2 m stick
Air Track
Air Cart with Springs
Triple Beam Balance
Spherical mass and string
Ultrasonic Motion Sensor
1.0
Introduction
Periodic motion, from masses on springs to vibrations of atoms, is one of the most important
kinds of physical behavior.
In this lab, we will deal with Hooke’s law, where the force is proportional to
the displacement, tending to restore objects to some equilibrium position. A large number of physics
systems can be successfully modeled with this simple idea, including the vibrations of strings, the
swinging of a pendulum, and the propagation of waves of all kinds. All of these phenomena involve
periodic motion.
Periodic vibrations can cause disturbances that move through a medium in the form of waves.
Many kinds of waves occur in nature, such as sound waves, water waves, seismic waves, and
electromagnetic waves. These very different physical phenomena are described by common terms and
concepts that will be introduced in this lab.
2.0
Theory
2.1
Simple Harmonic Motion
One important property of oscillator (periodic) motion is called its
frequency,
or number of
oscillations that are completed each second.
The symbol for frequency is
f,
and the SI unit is
hertz
(abbreviated Hz), where
1 hertz = 1 Hz = 1 oscillation per second = 1 s
1
(1)
Related to the frequency is the
period
T
of the motion, where period is the time for one complete
oscillation (or
cycle
); that is
ܶ ൌ
1
݂
(2)
Any motion that repeats itself at regular intervals is called
periodic motion
or
harmonic motion.
We are interested in the displacement
x
of a particle from the origin given as a function of time by
ݔሺݐሻ ൌ ݔ
cosሺ߱ݐ ߶ሻ
(3)
where
x
m
,
ω
,
and
φ
are constants. This motion is called
simple harmonic motion
(SHM), a term that
means the periodic motion is a sinusoidal function of time.
Equation (3) is the sinusoidal function of a
cosine function shown in Figure 1.
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Figure 1
A graph of equation (3) taken from
Wikipedia.
Figure 2
A handy reference to the quantities in equation
(3) for simple harmonic motion.
The quantity
x
m
, called the
amplitude
of the motion, is a positive constant whose value depends
on how the motion was started.
The subscript
m
stands for
maximum
because the amplitude is the
magnitude of the maximum displacement of the particle in either direction. The cosine function in
equation (3) varies between the limits ±1; so the displacement x(t)
varies between the limits ±x
m
.
The timevarying quantity
)
(
φ
ω
+
t
in equation (3) is called the
phase
of the motion, and the
constant
φ
is called the
phase constant
(or
phase angle)
.
The value of
φ
depends on the displacement
and velocity of the particle at time t=0.
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 Fall '09
 Mass, Simple Harmonic Motion, Rotary Motion Sensor

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