An analysis of Simple Harmonic Motion.
Simple Harmonic Motion.
If a particle moves in a straight line in such a way that its acceleration is always directed towards a fixed point on the line and is proportional to the distance from the point, the particle is said to be moving in Simple Harmonic Motion.
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Let O be a fixed point on the line X'X and x the distance of the particle from O at any time t. Also, let the acceleration of the particle along OX as
is a positive constant. No matter whether x is positive or negative the acceleration will be directed towards O.
If v is the velocity at time t, the acceleration will be in the direction OX and will be given by the differential relationship :-
This can be expressed as:
In the initial stage of the motion v is negative as the particle is moving towards O
When t = 0 x = a and
= 0 and hence K = 0
cos nt = 0, and thus a particle starting from A moving towards O arrives in a time
with a velocity of
. It will continue along the straight line and its velocity will be zero when
. It will then return to O arriving when
with a velocity
and reach A in a time
with zero velocity. The motion is then repeated indefinitely unless destroyed by some force.
- The time is called the Period of the oscillation and is the time for one complete cycle.
- If the frequency is f and the period , then .
- If the period of the motion is known, the motion is completely determined.
- The Period maybe written down at once if the magnitude of the acceleration for some value of x is known.
- The amplitude is determined by the initial displacement.
Other Initial Conditions
If the the motion is started by giving the particle a velocity
when its distance from O is
, the type of motion is unchanged and the time is measured from this instant, instead of the instant when x = a. In this case the value of x at any instant is given by :-
is a constant]
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is called the Epoch
of the motion. The Phase
of the motion at time t is the time which has elapsed since the particle was at the positive end of its path. Thus the phase is
less a multiple of the period.
In particular if
, i.e. the particle starts from O
and the amplitude is
Since the acceleration at any instant is
This is characteristic of Simple Harmonic Motion and its solution is given by:-
may be written down if the amplitude and the epoch are known.
The Relation To Uniform Motion In A Circle.
If a particle is describing a circle of radius
with uniform angular velocity
, its orthogonal projection on a diameter of the circle moves on the diameter in simple harmonic motion of amplitude
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be the angle which the radius initially makes with the diameter X'OX . Then after a time t the angle made by the radius to the particle is
. Hence, if P is the position of the particle at time t and N the foot of the perpendicular from P on OX, then:-
and the point moves with Simple Harmonic Motion of amplitude
Example - Simple Harmonic Motion
A particle moves with Simple Harmonic Motion in a straight line. Find the time of a complete
oscillation if the acceleration is 4 ft/sec2, when the distance from the centre of the
oscillation is 2 ft. If the Velocity with which the particle passes through the centre of
oscillations is 8 ft./sec. find the amplitude.
If the acceleration is
at a distance x
from the centre then:
Hence the period is:
If the phase is zero when
is the amplitude. Then
And the value of v
at the centre of oscillation is
The period is