CONTENTS
àIntroduction
àReview of Literature
àTheory
àSummary
àBibliography
Introduction
Simple harmonic motion is the motion of a simple harmonic oscillator, a motion that is neither driven nor damped. The motion is periodic, as it repeats itself at standard intervals in a specific manner - described as being sinusoidal, with constant amplitude. It is characterized by its amplitude (which is always positive), its period which is the time for a single oscillation, its frequency which is the number of cycles per unit time, and its phase, which determines the starting point on the sine wave.
Simple harmonic motion is defined by the differential equation, where "k" is a positive constant, "m" is the mass of the body, and "x" is its displacement from the mean position.
In words, simple harmonic motion is "motion where the force acting on a body and thereby acceleration of the body is proportional to, and opposite in direction to the displacement from its equilibrium position" (i.e. F = − kx ).
A general equation describing simple harmonic motion is where x is the displacement, A is the amplitude of oscillation, f is the frequency, t is the elapsed time, and φ is the phase of oscillation. If there is no displacement at time t = 0, the phase. A motion with frequency f has period, Simple harmonic motion can serve as a mathematical model of a variety of motions and provides the basis of the characterization of more complicated motions through the techniques of Fourier analysis.
Review of Literature
ð
2007
www.physicsforums.com
The Swing Analog : Velocity Gradation in SHM
In Simple Harmonic Motion, the movement does not stop with the elastic recoil of the tines. That is, the tines of the tuning fork continue to move through their rest position because of inertia, the tendency for motion or lack of motion to continue. The graph of body vibrating in simple harmonic motion is same as the graph of a sine wave. The pattern is simple because there is only one frequency of vibration. The movement repeats itself until it damps out & therefore it is periodic.
ð Ertan Salik
1989
www.physicslab.org
Why is simple harmonic motion 'simple'?
If you look at a text on Simple Harmonic Motion in a physics book you see that 'Simple' refers to the ideal case where there is no friction, viscosity etc. Indeed, ideal cases are usually the simples in Physics. But many books also have parts on 'Damped Oscillations' and 'Forced Oscillations' but these are not considered as simple, because they are closer to real cases. Also the solutions to ideal case is the simplest, and the solutions to forced and damped oscillations are more complicated as one could expect.
ð Raymond A Servey , John W Jewett
2006
www.phys-astro.sonoma.edu
SHM: Oscillator
If we consider SHM, we have a natural time scale or clock that describes the timing of the motion. If we look at the system with a strobe light that then the oscillator appears as a stationary point, we find that when the period of oscillator is perturbed, we must readjust or stroboscopic device in order to maintain our view of the system as a fixed point.
ð Edward Leamingtor Nicholas
(American institute of physic, Cornell university)
1896
www.hyperphysics,phy-astr.gsu.edu
Damped simple harmonic motion
Oscillatory motion can be described by a period, which tells you how long it takes per oscillation, and a frequency, which is how many oscillations per second. The size of the oscillations is described by the amplitude. When a mass on a spring oscillates it has kinetic energy and elastic potential energy and gravitational potential energy. The total energy is the sum of these and is conserved. However, we know that if we start a spring oscillating, it will eventually stop, because of friction. This is known as damped simple harmonic motion.
To keep the spring oscillating we need to provide a driving force.
ð Peter B Kahn
2004
www.explorescience.com
Simple Harmonic Motion
Simple harmonic motion is characterized as periodic motion with constant amplitude & constant frequency. This means that after each period the amplitude returns to its initial value. The frequency is independent of the amplitude .So we conclude that two identical simple harmonic oscillators that have differing initial conditions still have the same period of oscillation.
Theory
We consider only one-dimensional oscillations (motion along a straight line). The oscillation frequency (v, vibrations per second) and period (T, seconds) are both related to the “circular frequency”, ω:
T = 2π/ω and v =ω/2π
What determines ω is the relation between the magnitudes of the strength of the restoring force and the inertia in the system,
ω=k/m , where Fnet=-kx
is the resultant force acting on the mass m when it is at position x, as measured from the equilibrium position. A resultant force obeying this equation (“elastic restoring force”) is necessary and sufficient to generate simple harmonic motion. Thus ω is an intrinsic property of the vibrating system and does not depend on the “initial conditions” (the position and velocity with which the vibration is started off). Since the vibration amplitude is determined by the initial conditions, it follows that ω is independent of amplitude for simple harmonic motion.
Vertical oscillation of mass on spring
The mass moves under the influence of the force of gravity and the spring force,
Fspring = - k (L - L0)
if L0 is the spring’s relaxed length. When the spring is hung from its upper end with M attached to the bottom, the equilibrium length is L1, determined by and hence the resultant force acting on M is Mg = K(L1- L0) and hence the resultant force acting on M is
Fnet = Fspring – Mg = - kx , where x = L – L1
All the L values are positive, but x may be positive or negative, with the direction of Fnet being opposite to the direction of x. Thus the “k” which determined ω in this case is the force constant of the spring itself; the force of gravity sets the equilibrium position but does not affect the vibration rate. This simplified analysis assumes that the spring has no mass. The real spring does have mass, of course, which also takes part in the oscillation. The part of the spring to which the M is attached vibrates with the same amplitude as M itself; the part of the spring attached to the upper end does not vibrate at all; and intermediate parts oscillate with intermediate amplitudes. The effect of this partial (on the frequency) is as if a certain fraction of the spring’s mass was added to the attached mass and the spring itself was massless.
Velocity in SHM
In simple harmonic motion, the velocity constantly changes, oscillating just as the displacement does. When the displacement is maximum, however, the velocity is zero; when the displacement is zero, the velocity is maximum. It turns out that the velocity is given by:
Acceleration in SHM
The acceleration also oscillates in simple harmonic motion. If you consider a mass on a spring, when the displacement is zero the acceleration is also zero, because the spring applies no force. When the displacement is maximum, the acceleration is maximum, because the spring applies maximum force; the force applied by the spring is in the opposite direction as the displacement. The acceleration is given by:
Note that the equation for acceleration is similar to the equation for displacement. The acceleration can in fact be written as:
All of the equations above, for displacement, velocity, and acceleration as a function of time, apply to any system undergoing simple harmonic motion. What distinguishes one system from another is what determines the frequency of the motion. We'll look at that for two systems, a mass on a spring, and a pendulum.
The frequency of the motion for a mass on a spring
For SHM, the oscillation frequency depends on the restoring force. For a mass on a spring, where the restoring force is
F = -kx, this gives:
This is the net force acting, so it equals ma:
This gives a relationship between the angular velocity, the spring constant, and the mass:
Summary
Simple harmonic motion (SHM) occurs when a system in stable equilibrium is “tweaked” (i.e. displaced slightly from equilibrium and released). Consider, for example, a system composed of a mass attached to a spring as shown to the right. When the system is at equilibrium, it is stable. If the mass is pushed to the left, the spring will push the mass to the right; if the mass is pulled to the right, the spring will pull the mass to the left. The force the spring exerts is referred to as the restoring force and follows Hooke’s Law. It is called restoring force because it pushes or pulls the system back into equilibrium. We refer to the restoring force in this case as a linear restoring force because, by Hooke’s Law (F = kx), the magnitude of this force is directly proportional to the distance by which the system is moved from equilibrium. Why all this attention to restoring force? Any system that can undergo simple harmonic motion is governed by a linear restoring force.
Bibliography
Sites:-
www.explorescience.com
www.encyclopedis.com
www.physicsforums.com
dev.physicslab.org
www.wikipedia.com
Books:-
Physics: Super Review
Fundamentals of Physics by Halliday, Resnick &
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