Simple Harmonic Motion
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. 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 :-
- 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 ConditionsIf 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 :-
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 and period . Let 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:-
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
The period is