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15.1. Simple Harmonic Motion
Any motion that repeats itself at regular intervals is called
harmonic motion
. A particle
experiences a
simple harmonics motion
if its displacement from the origin as function of time
is given by
where x
m
, [omega] and [phi] are constants, independent of time. The quantity x
m
is called the
amplitude of the motion
and is the maximum displacement of the mass. The time-varying
quantity ([omega]t + [phi]) is called the
phase of the motion
and [phi] is called the
phase
constant
. The phase constant is determined by the initial conditions. The
angular frequency
[omega] is a characteristic of the system, and does not depend on the initial conditions. The unit
of angular frequency is rad/s. The
period
T of the motion is defined as the time required to
complete one oscillation. Therefore, the displacement x(t) must return to its initial value after one
period
x(t) = x(t + T)
This is equivalent to
Using the relation
it is immediately clear that
The number of oscillations carried out per second is called the
frequency of the oscillation
. The
symbol for frequency is [nu] and its unit is the Hertz (Hz):
1 Hz = 1 oscillation per second = 1 s
-1
The period T and the frequency [nu] are related as follows
The velocity of an object carrying out simple harmonic motion can be calculated easily
The positive quantity [omega] x
m
is called the
velocity amplitude
and is the maximum velocity
of the object. Note that the phase of the velocity and displacement differ by 90deg.
. This means
that
the velocity is greatest when the displacement is zero and vice versa
. The acceleration of
an object carrying out simple harmonic motion is given by
The positive quantity [omega]
2
x
m
is the
acceleration amplitude
a
m
. Using the expression for
x(t), the expression for a(t) can be rewritten as
This shows that the acceleration is proportional to the displacement, but opposite in sign. The
force acting on the mass can be calculated using Newton's second law
This equation of force is similar to the force exerted by a spring (Hooke's law)

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