Harmonic Motion Notes

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All periodic motions are not simple harmonic. The characteristics of simple harmonic motion notes are the motion of a body is in a straight line to and fro about a fixed point. This fixed point is called the equilibrium position. The force acting on a body is always directed towards fixed point and its directly proportional to its displacement from that point.

Equation of Motion in Simple Harmonic Motion

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If displacement of body from fixed point is y and the force on the body is F, then for simple harmonic motion

This is necessary and sufficient condition for simple harmonic motion is that the restoring force (F) is proportional to displacement(y).

F ∞ - y or F = - ky

Where k is a constant.

The acceleration ‘a’ of particle is given by

As acceleration is rate of change of velocity of particle

Integrating, we get

Now at y = y0, v = v0 (say); then equation (4) gives

In view of this equation (4) gives

General Equation

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The general equation ...
... of simple harmonic motion is y=A sin (ωt + φ) where (ωt + φ) is phase of particle at time t.

Where φ is constant of integration.

This is equation of motion of simple harmonic motion notes.

Projection of Uniform Circular Motion

Check this Common Physics Formulas awesome i recently used to see.

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Consider a particle of mass m, to be rotating with uniform angular velocity ω in a circular path of radius A. clearly the motion of the particle is periodic, but not simple harmonic. When the particle is at P, then the foot of perpendicular PN drawn from P on diameter YY’ of circle is at point N, when particle is at Y, the foot of perpendicular is also at Y, when particle reaches X’, the foot of perpendicular moves from y to O, when particle reaches Y’, the foot of particle moves to Y’-when particle is at X, the foot reaches to point O. when particle returns again to P, the foot again reaches N.

What Is Damped Simple Harmonic Motion
When a spring is stretched a distance x from its equilibrium position, according to Hooke's law it exerts a restoring force F = - kx where the constant k is called the spring constant. If the spring is attached to a mass m, then by Newton's second law, −kx = m x .. , where x .. is the second derivative of x with respect to time. This differential equation has the familiar solution for oscillatory (simple harmonic) motion: x = Acos( ωt + φ), (1) where A and φ are constants determined by the initial conditions and ω= k / m is the angular frequency. The period is T = 2 π m k . By differentiating Eq.(1) we determine the velocity v = −A ωsin(ωt + φ), which can be rewritten as v = Aωcos(ωt + π 2 + φ). (2) By differentiating again we obtain the acceleration a = −Aω2 cos( ωt + φ), which can be rewritten as a = Aω2 cos( ωt + π+ φ). (3) From the acceleration we find the force, F = mAω2 cos( ωt + π+ φ). (4) From these four equations we see that the velocity leads the displacement in phase by π/2 while the force and acceleration lead by π. For actual oscillating masses the motion is frequently not quite this simple since frictional forces act to retard the motion.

Simple harmonic motion is a type of motion which repeats itself after a specific interval of time. It is a repetitive motion in which the body keeps following a specific pattern of motion, in simple harmonic motion the object covers some distance about a central position called the mean position. The point at which the object reaches after covering the maximum distance is called the extreme position. The distance between the maximum position and the mean position is called amplitude of the motion. However once this to and fro motion is started the body will always not touch the maximum or extreme position in reality its distance will start decreasing gradually. This simple harmonic motion is found in several objects, some very fine examples are the motion of pendulum and movements of an object tied to a spring. It is assumed that when any object with its one end tied to a fixed point is displaced from its position it can execute simple harmonic motion, and there is always some work done to displace the object initially which is then used to keep the object moving. Damped motion on the other hand is a motion in which an attempt is made to reduce the amplitude of the vibrating body. So a damped simple harmonic motion is the motion in an object tied to a fixed point is displaced from its position, and it will start executing simpler harmonic motion. Once the motion is started through some external source or through the inheritance of the system the amplitude of the motion is damped.

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