Periodic Motion
Definition of Terms
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Periodic Motion: Motion that repeats itself in a
regular pattern.
Cycle: One complete vibration or oscillation.
Equilibrium position: Position of the object
when it is at rest. No energy is stored, and no
net force acts on the object.
Restoring Force: Force that brings an
oscillating object back to its equilibrium position.
Robert Hooke (1635-1703)
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In 1678, he
determined that the
deformation of an
elastic object is
directly proportional to
the force causing the
deformation.
(HOOKE’S LAW)
Hooke’s Law
Fx
or
F = -kx
F = applied force (N)
 x = amount of deformation (m)
 k = spring (force) constant
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Units = N/m
Ex: A 0.25 kg mass is suspended from a
spring with a force constant of 48 N/m. How
far does the spring stretch?
Given: m = 0.25 kg
F = mg = -2.4 N
k = 48 N/m
Find: x = ?
F = -kx
F/k = x
(2.4 N) / (48 N/m) = x
0.050 m = x
Elastic Potential Energy (PEe)
Energy stored in an elastic object (usually
a spring) by deforming it (doing work on it).
PEe = ½ kx2
 x = distance spring is deformed (stretched
or compressed)
 k = spring constant: How resistant an
elastic object is to being stretched or
compressed (stiffness). Units = N/m
 Units = N/m (m2) = N*m = Joules
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Ex: A spring with a spring constant of
160 N/m is compressed by 8.0 cm.
How much energy is stored in the spring?
Given: k = 160 N/m
x = 0.080 m
PEe = ½ kx2
= ½ (160 N/m)(.080 m)2
PEe = 0.51 J
Find: PEe = ?
Simple Harmonic Motion (SHM)
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Any periodic motion in which the restoring
force is proportional to the displacement
from equilibrium.
Mass on a Spring
 Simple Pendulum (for small angles)
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Measures of Simple Harmonic
Motion
Amplitude (A): Maximum displacement
from equilibrium position (meters)
 Period (T): Time it takes to execute one
complete cycle of motion (seconds)
 Frequency (f): Number of cycles or
vibrations per unit of time
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Units = Hertz (Hz) or s-1 or 1/s
Calculating Period
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Simple Pendulum:
L
T = 2π
g
T = period (s)
L = length of pendulum (m)
g = free fall acceleration (m/s2)
Ex: What is the period on Earth of a simple
pendulum that has a length of 1.20 m?
Given: L = 1.20 m
g = 9.81 m/s2
Find: T =?
T = 2π√(L/g)
= 2π√(1.20 m / 9.81
m/s2)
= 2.20 s
Resonance
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Forced Vibrations: One vibrating object causes
another object to vibrate at the same frequency.
Natural Frequency: Frequency at which
minimum energy is required to produce forced
vibrations. An object “prefers” to vibrate at this
frequency.
Resonance occurs when the frequency of a
forced vibration matches the natural frequency
of a system.