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  Simple Harmonic Motion-I, Physics tutorial

Introduction:

Simple harmonic motion,  in physics, repetitive movement backward and forward through the equilibrium, or central, position, so that maximum displacement on one side of the position is equal to maximum displacement on other side. Time interval of every complete vibration is same, and force liable for motion is always directed toward equilibrium position and is directly proportional to distance from it.

Several physical systems show simple harmonic motion (suppose no energy loss): oscillating pendulum, electrons in the wire carrying alternating current, vibrating particles of medium in the sound wave, and other assemblages comprising relatively small oscillations about the position of stable equilibrium.

Particular example of the simple harmonic oscillator is vibration of the mass joined to the vertical spring, other end of which is fixed in the ceiling. At maximum displacement -x, spring is under the greatest tension that forces mass upward. At maximum displacement +x, spring reaches the greatest compression that forces mass back downward again. At either position of maximum displacement, force is greatest and is directed toward equilibrium position, the velocity (v) of mass is zero, the acceleration is at the maximum, and mass changes direction. At equilibrium position, velocity is at its maximum and acceleration (a) has fallen to zero. Simple harmonic motion is classified by changing acceleration which always is directed toward equilibrium position and is proportional to displacement from equilibrium position. Also, interval of time for every complete vibration is constant and doesn't rely on size of maximum displacement. In some form, thus, simple harmonic motion is at heart of timekeeping.

Equation relating acceleration and displacement in a s.h.m. is

a ∝ x

Therefore a = (- constant) x

Negative sign indicates that acceleration is always in opposite direction to displacement and directed to the fixed point.

Relating S.H.M. with Circular Motion:

533_Relating SHM with Circular Motion.jpg

In the given figure consider point P move round circle of radius r and centre O with uniform angular velocity ω. It will have the constant speed V round circumference. Speed V is equal to ωr. As P moves round circle in the direction shown (i.e. anti clockwise), N the foot of perpendicular from P on the diameter AOB moves from A to O to B and back to A through O. By the time N comes back to point A, P also completes one cycle. Now, consider initial positions of N and P be at A at time t =O. At a later time, t = t, N and P are now as pointed out in diagram with radius OP making angle O with OA. Consider distance ON is x. Below some parameters are described that govern s.h.m. to show that motion of N from A to B and back to A is simple harmonic about O.

Acceleration:

The motion of N is because of that of P therefore acceleration of N is component of acceleration of P parallel to AB. The acceleration of P is ω2r (or v2/r) along PO. Therefore component of this parallel to AB is simply ω2rn Cos θ. thus acceleration a of N is a = -ω2rcosΘ

Negative sign, illustrates mathematically that acceleration is always directed towards O.

But, x = r Cos Θ in the diagram

Therefore a = -ω2x

This equation defines that acceleration of N towards O is directly proportional to distance from O. We conclude that N explains a s.h.m. about O as P revolves round circle-called auxiliary circle -with constant speed.

Period:

Period T of N is time it takes N to do one complete to and fro motion i.e. to go from one point to another and back to initial point. In the same time, P will move round auxiliary circles once. Therefore,

T = Circumference of Auxiliary circle/speed of P

but V = wr

Therefore T = 2Πr/v= 2Π/w

For the particular s.h.m ω is constant and so T is constant and independent of amplitude r of oscillation. If amplitude increases, body travels faster and so T remains unchanged. Motion that, has constant period whatever amplitude, is said to be isochronous. This property is the significant feature of s.h.m. Frequency f is number of complete oscillations per unit time. I.e. f =I/T. The oscillation per second is a hertz.

Velocity:

Velocity of N is the same as the component of P's is velocity parallel to AB which

= -v sin Θ

=-ωr sin Θ

As sin Θ is positive when 0o< Θ < 180o, that is, N moving upwards, and negative when 180o<Θ<360+, i.e. N moving downwards, negative sign makes sure acting upwards and positive when acting downwards. Variation of velocity of N with time (suppose P, and so N, start from A at time zero)

=-ωr sin ωt(since Θ = ωt)

Variation of velocity of N with displacement

X = -ωr sinΘ

= ±ωr√(1-cos2Θ)

= ±ωr√(1-(x/r)2)

= ±ω√(r2-x2)

Therefore velocity of N is

± ωr (a maximum) when x = O

zero when x = ±r

Displacement:

This is provided by:

x = r Cos θ = r Cos ωt

Maximum displacement OA or OB is known as amplitude of oscillation. Graph of variation of displacement of N with time is shown in Figure. It is a sinusoidal pattern just as graphs of velocity and acceleration with time.

1018_sinusoidal pattern.jpg

Examine that when velocity is zero, acceleration is the maximum and vice versa. There exists a phase difference of a quarter of a period (i.e. T/4) between velocity and acceleration.

Expression for ω:

Now find what quantity ω is equal to in a s.h.m.

a = -ω2r

Ignoring sign we can write

ω2 = a/x= ma/mx = (ma/x)m

Where m is mass of the system.

Force causing the acceleration a at displacement x is ma, thus ma/x is force per unit displacement. Therefore,

ω = √[(force per unit displacement)/(mass of ocillating system)]

Period T of the s.h.m is provided by

T = 2Π/ω

= 2Π√(mass of oscillating system)(force per unit displacement)

This expression tells that T increases if (i) mass of oscillating system increases and (ii) force per unit displacement decreases i.e. if elasticity factor decreases.

A vibration is simple harmonic if it's equation of motion can be written in the form

a = - (positive constant) x

And, by convention, signify this positive constant by ω2 as T=2π/ω. Hence, ω is square root of positive constant in the acceleration -displacement equation.

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