Definition of Simple Harmonic Motion:
Simple Harmonic Motion (or simply SHM) is the periodic motion of a body or particle all along a straight line in such a manner that the acceleration of the body is directed towards a fixed point (or centre of motion) and is as well proportional to its displacement from that point.
Illustration of Simple Harmonic Motion:
Illustration of Simple harmonic motion comprise the:
Mathematical representation of SHM:
The main characteristics of SHM are that the motion is periodic or repetitive and that it comprises a maximum displacement from the position of the equilibrium. Such characteristics should appear in any mathematical demonstration of SHM.
The magnitude of the highest displacement from equilibrium is termed as the amplitude A of the motion. As a magnitude, the amplitude can't be negative. The period of time over which the motion repeats is termed as the period or periodic time T and this can be computed from any point in the motion to the subsequent equivalent point. One complete period is just adequate for one complete cycle or oscillation of the motion, from a turning point, state, via the position of equilibrium to the other turning point and back again to the original turning point. The frequency 'f' is the number of cycles completed in one second and is the reciprocal of the period (that is, evaluated in seconds):
f = 1/T
The units of frequency are employed so frequently and so broadly that they have been given the special term hertz, which might be abbreviated to Hz. Therefore,
1 hertz = 1 Hz = 1 second-1 = 1 s-1
When the moving object travels forward and backward and all along a straight line (illustration: suspending spring having mass) we state the motion is linear and refer to it as one-dimensional SHM. In such conditions we can take the place of equilibrium to be the origin of a Cartesian coordinate system and by selecting the orientation of the system suitably we can symbolize the displacement from equilibrium, measured all along the line of motion, through the position coordinate x.
To signify the variation of displacement by means of time mathematically we need x as a Periodic function of time 't'. The simplest periodic functions to employ are sine and cosine, therefore we could, for illustration, represent a one-dimensional periodic motion, regarding the origin by:
x(t) = A cos [(2πt)/T]
Here, A and T are constants, and x(t) signifies the displacement from the position of equilibrium at time t.
The figure above just represents over two cycles of this linear motion as a displacement-time graph. Observe the graph carefully, and note that, in all necessary details, it corresponds very much to the qualitative explanation of SHM given earlier. In specific, note that:
a) The motion is periodic by periodic time T. Whenever you compare the part of above figure from t = 0 to t = T with that from 1 = T to t = 2T, such parts are evidently similarly. The graph therefore repeats itself after a time T.
b) It consists of a maximum displacement or amplitude A.
c) It represents an object moving smoothly and symmetrically among the two turning points.
d) It represents an object that is stationary at the turning points and that moves by a changing velocity vx among the turning points, achieving a maximum speed as it passes via the position of equilibrium at x = 0.
e) As the velocity vx modifies smoothly among the turning points the acceleration should as well be changing smoothly over a cycle and, from Newton's second law of motion Fx = max, this means a smoothly varying force acting on the object.
Uniform circular motion and SHM:
The Greek letter ω which is employed to stand for the angular frequency of an oscillator is as well broadly employed to symbolize the angular speed of an object moving in the circle. This is no mere coincidence, instead it points to a deep relationship among one-dimensional simple harmonic motion and uniform circular motion. (An illustration of uniform circular motion would be the movement of the point on the rim of a uniformly rotating wheel of radius r. The direction of motion of such a point is modifying all the time however its speed u is constant, and therefore is its angular speed ω = v/r.) This link between SHM and uniform circular motion gives valuable insight into the ideas of phase and angular frequency.
The figure above represents an object in uniform circular motion regarding the origin of a Cartesian coordinate system. (The x-axis of the coordinate system is as well shown in the above figure.) The object is moving having constant angular speed ω at a fixed distance A from 0, so its speed at any instant is provided by v = Aω
However, the apparent linear motion observe from the plane of figure above would be SHM having amplitude A (that is, equivalent to the radius of circle) and angular frequency ω (that is, equivalent to the angular speed of the circular motion). Therefore by viewing the uniform circular motion in the right manner we can get the appearance of one-dimensional SHM. Or, to put it other manner, by any given one-dimensional SHM of amplitude A and angular frequency ω we can relate a fictional uniform circular motion of radius A and angular speed ω.
Now, the relationship between one-dimensional SHM and uniform circular motion assist us to comprehend the phase which appears in the mathematical presentation of the SHM,
x = A cos ωt
x = A sin ωt
Now, assume that we start timing the motion in the figure above from the moment if the moving object crosses the x-axis at point X. When the time taken for the object to reach point P is t then we can state that θ = ωt, where θ is the angle between the line OP and the x-axis.
If we now suppose that viewing this motion from the plane of figure above and in a direction perpendicular to the x-axis, in such a way that what we notice appears to be one-dimensional SHM all along the x-axis, then it is quite obvious that the displacement of the moving object from the centre of the motion will be provided by x = OQ = A cos θ = A cos ωt, that is,
x = A cost ωt
Therefore, given any illustration of SHM Mar can be explained by equation above, we can recognize the phase on at any time t by the change in angular position θ of the moving object in the related uniform circular motion. Moreover, we can state that initially (if t = 0) the oscillator is at x = A and θ = 0.
Now assume that we notice the similar circular motion again, however this time we begin timing from the earlier moment if the moving object is directly beneath the origin in the figure shown above. This signifies that by the time the moving object arrives at the point X it will already have moved via an angle of Π/2 radians (that is, 90°). Under such conditions the angular position of the moving object after a time t will be θ = ωt - Π/2 rad, where θ is still being computed anticlockwise from the x-axis.
Velocity and acceleration in SHM:
We are familiar that the gradient of the displacement-time graph at any time provides the velocity of the moving object at that time; the velocity-time graph of the moving object can therefore be made from the displacement-time graph by computing the gradient dx/dt of the latter graph at each and every point and through plotting this gradient as a function of time. The acceleration-time graph can then is made from the gradient dvx/dt of the velocity-time graph in an identical manner. This graphical method is both slow and imprecise.
Luckily there is a quick and precise alternative process for determining the gradient of a graph at any point, provided we know the algebraic expression which explains the curve, that is, provided we are familiar with the function concerned. This process employs the methods of differential calculus to determine the gradient of the graph. When the displacement x at time t is given by the function x(t) then the gradient of the displacement-time graph at time t is provided by dx(t)/dt and this is as well the instantaneous velocity vx(t) of the object at that time. In an identical manner the gradient of the velocity-time graph at time t is given by dvx(t)/dt and this is as well the instantaneous acceleration ax(t) of the object at time t. We can summarize such statements as follows:
vx(t) = (dx/dt)(t)
ax(t) = (dvx/dt)(t) = (d2x/dt2) (t)
Therefore, beginning from our selected explanation of SHM, if we now replace,
x(t) = A cos (ωt + Φ)
into the given Equations:
vx(t) = (dx/dt)(t)
ax(t)= (dvx/dt)(t) = (d2x/dt2) (t)
And preceding either graphically or through calculus, we find out:
vx(t) = - A ω sin(ωt + Φ)
and ax(t)= - A ω2 cos(ωt + Φ)
Comparing the equations:
ax(t) = - A ω2 cos (ωt + Φ) = - ω2x(t)
Forces acting in SHM:
The force acting on an object is associated to the acceleration of the object via Newton's second law of motion, F = ma.
Equation ax(t) = - ω2x(t), provides the acceleration at time t in terms of the displacement at time t. We can utilize this outcome to infer the force acting in SHM. When the object consists of mass m then the x-component of the force acting at time t should be given by the expression:
Fx(t)= max(t) = - mω2x(t)
The importance of equation is:
In SHM, the magnitude of the force acting at any time is linearly proportional to the distance from equilibrium at time and the direction of the force is just opposite to that of the displacement, that is, towards the equilibrium position.
Therefore the force causing SHM for all time acts in a direction which tends to decrease the displacement and for this reason it is generally termed as the restoring force.
Fx(t) = max(t) = - mω2x(t)
The above provides us a general expression for the restoring force acting on the SHM oscillator, in terms of mass and angular frequency of oscillation. It means that the force responsible for SHM is linearly proportional to the displacement x from the position of equilibrium and is for all time directed towards it. We could put this in a more enlightening manner by stating:
SHM will take place if an object moves from equilibrium beneath a restoring force that is linearly proportional to the displacement from equilibrium-which is it moves beneath the influence of a linear restoring force.
The constant of proportionality in Fx(t) = max(t) = - mω2x(t); k = mω2, is termed as the force constant for the motion.
k = - Fx(t)/x(t) = mω2
Therefore, ω = √(k/m)
and the period of the oscillation is provided by:
T = 1/f = 2 Π/ω
T = 1/f = 2 Π√(m/k)
Superposition of SHMs:
We now consider what happens in the common condition where the motion of an object outcomes from the combination of some independent SHM oscillations at similar time. At times such independent oscillations comprise collinear displacements (that is, all along the similar line) and at times they are in dissimilar directions. An illustration of the former would be the in and out vibrations of the cone of a loudspeaker, if driven through an amplifier. An illustration of the latter would be the motion of a ship in a rough sea - side to side, up and down and stem to prow.
The other illustration of the addition of displacements in dissimilar directions is the motion of an atom in a crystal. Each and every atom experiences a force from neighboring atoms. Such forces encompass attractive and repulsive features and so act instead similar to springs joining the atoms altogether. We can model this by imagining each atom to be joined to three springs, one all along each of the three coordinate axes x, y and z of the crystal. A specific atom will then oscillate about a place of equilibrium beneath the joint effect of three mutually perpendicular motions all along each of such axes. This three-dimensional oscillation sounds dreadfully complex however it is in reality surprisingly simple to deal with in terms of the addition of three independent one-dimensional motions in x, y and z.
The general word to explain the addition of two or more oscillations is the superposition. The superposition principle merely states that:
If several oscillations are added, the resultant displacement at any time is the sum of the displacements due to each and every oscillation at that time.
The superposition principle is valid merely for linear systems as it rests on the linear response of the system to the combined oscillation, and thus all displacements should remain adequately small. Superposition of the oscillations in linear systems is the subject of this part and it comprises the application of a few trigonometries.
Superposition of collinear SHMs: beating and beat frequency
Let take the superposition of two SHM displacements all along the x-axis:
x(t) = xl(t) + x2(t)
In common, the two SHMs might encompass various amplitudes, angular frequencies and phase constants so we can use:
x(t) = A cos[(2Πt)/T + Φ)] = A cos(ωt + Φ)
To write this as:
x(t)= A1 cos (ω1t + Φ1) + A2 cos (ω2t + Φ2)
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