Transcript Simple Harmonic Motion

Simple Harmonic Motion
• Simple harmonic motion (SHM) refers to a
certain kind of oscillatory, or wave-like motion
that describes the behavior of many physical
phenomena:
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a pendulum
a bob attached to a spring
low amplitude waves in air (sound), water, the ground
the electromagnetic field of laser light
vibration of a plucked guitar string
the electric current of most AC power supplies
SHM Position, Velocity, and Acceleration
Simple Harmonic Motion
Periodic Motion: any motion of system
which repeats itself at regular, equal
intervals of time.
Simple Harmonic Motion
Simple Harmonic Motion
• Equilibrium: the position at which no net force acts on
the particle.
• Displacement: The distance of the particle from its
equilibrium position. Usually denoted as x(t) with x=0 as
the equilibrium position.
• Amplitude: the maximum value of the displacement with
out regard to sign. Denoted as xmax or A.
The period and frequency of a
wave
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the period T of a wave is the amount of time it takes to
go through 1 cycle
the frequency f is the number of cycles per second
– the unit of a cycle-per-second is commonly referred to as
a hertz (Hz), after Heinrich Hertz (1847-1894), who
discovered radio waves.
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frequency and period are related as follows:
1
f 
T
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t
Since a cycle is 2p radians, the relationship between
frequency and angular frequency is:
  2pf
T
Here is a ball moving back and forth with simple
harmonic motion (SHM):
Its position x as a function of time t is:
where A is the amplitude of motion : the distance from the
centre of motion to either extreme
T is the period of motion: the time for one complete cycle
of the motion.
Springs and SHM
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Attach an object of mass m to the end of a spring, pull it out to a
distance A, and let it go from rest. The object will then undergo simple
harmonic motion:
x (t )  A cos(t )
v (t )   A sin( t )
a(t )   A 2 cos(t )
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What is the angular frequency in this case?
– Use Newton’s 2nd law, together with Hooke’s law, and the
above description of the acceleration to find:
k

m
Spring Constant, K
The constant k is called the spring
constant.
SI unit of k = N/m.
HOOKE'S LAW
The restoring force of an ideal spring is given by,
where k is the spring constant and x is the
displacement of the spring from its
unstrained length. The minus sign indicates
that the restoring force always points in a
direction opposite to the displacement of
the spring.
Simple Harmonic Motion
When there is a restoring force, F = -kx, simple harmonic
motion occurs.
Position VS. Time graph
Amplitude
Amplitude is the magnitude of the maximum displacement.
Period, T
For any object in simple harmonic motion, the time
required to complete one cycle is the period T.
Frequency, f
The frequency f of the simple harmonic motion is the
number of cycles of the motion per second.
Oscillating Mass
Consider a mass m attached to the end of a spring as
shown.
If the mass is pulled down and released, it will
undergo simple harmonic motion.
The period depends on the spring constant, k and
the mass m, as given below,
m
T  2p
.
k
2
T k
m
2
4p