Idea Transcript
Harmonic Motion Lore of wide use. Enjoy the great illustrative power of mathematics!
Why do sinusoidal variations with time seem to crop up everywhere in Physics? By sinusoidal I mean any variation that is a linear combination A sin 2ft + B cos 2ft, where A and B are constants, and f (sometimes )is the frequency in per second, or hertz (Hz). The angular frequency = 2f, in radians per second, is often convenient. The unit Hz should not be used with angular frequency, to avoid confusion. Our sinusoidal variation can also be written C cos (t + ), where C = (A 2 + B2)1/2 is the amplitude and = tan -1(A/B) is the phase constant. The quantity (t + ) is the phase. A variation of this type is called harmonic, or, more precisely, simple harmonic, since it involves only one frequency f. The Figure at the right shows a graphic representation of simple harmonic motion. The vector of length A rotates anticlockwise from its initial position at time 0 as shown by the dotted vector. The projection of the terminus of the rotating vector on the x-axis represents a simple harmonic motion (as does the projection on the y-axis). A vector (x,y) can also be expressed as a complex number in the form x + iy = Aeit = A (cos t + i sin t). It is very convenient to express a harmonic variation in complex exponential form, Aei(t + ) , because then all the rules of manipulation of exponents are available. The complex constant Aei is called a phasor, and is very useful for the specification of a harmonic variation. Engineers use j instead of i for the imaginary unit to avoid confusion with the symbol for electrical current. Physicists often take the time variation as e-it . In any case, two parameters are necessary for the complete specification of a simple harmonic motion. With these definitions out of the way, we can return to the question. The small-amplitude natural vibrations of bodies are observed to be simple harmonic, or else a sum of several simple harmonic variations. Ohm's Law (of acoustics) asserts that the fundamental aural perception is that of a simple harmonic variation. In fact, this is the origin of the word. Fourier's Theorem shows that an arbitrary periodic time variation can be expressed as a sum of simple harmonic variations, usually an infinite number. These are only a few examples of the ubiquity of harmonic motion. They all are consequences of the mathematical behaviour of simple harmonic motion, often combined with the principle of superposition. The two essential mathematical properties of simple harmonic motion are: (1)the sum of any number of such motions is also a harmonic motion of the same frequency, with at most a difference of amplitude and phase constant, and (2) the derivative (or integral) of a harmonic motion is also a harmonic motion of the same frequency, again with at most a difference of amplitude and phase constant. The exponential function eax also has these properties, but increases without bound for large positive or negative x (depending on the sign of a), so it is unsuitable for the description of a steady motion. The harmonic vibration, on the other hand, is periodic with period T = 1/f. Also, the two linearly independent functions e+ax and e-ax cannot be turned into one another by differentiation, as the sine and cosine can. The exponential and trigonometric functions are closely related, as we know, and the sine and cosine are linear combinations of complex exponentials. The differential equation d 2y/dx 2 + a2y = 0 has solutions e-iax and e+iax, or sin ax and cos ax, as can be found by substitution in the equation. Each pair of solutions is such that no linear combination of them is a constant, so they are called linearly independent. The general solution of the differential equation is y(x) = A sin ax + B cos ax, and it can be made to satisfy any two algebraic conditions, such as having specified values at two different points, or for having a specified value and derivative at a certain point. These conditions are variously called initial or boundary conditions. For example, B = y(0) and A = y'(0)/a, if the initial position and slope are specified at the origin. This differential equation occurs over and over, so it is good to remember its solutions and their properties. For example, consider a mass M hanging from a fixed point O by a flexible, inextensible, weightless cord of length L. The motion can be described in terms of the distance x that M moves to the right or left of its equilibrium position. When it is displaced by x, the weight Mg acts directly downward, but the force of the cord on M is angled a bit. The net horizontal force on M is then -Mg(x/L), if we assume that the displacement x is always small compared to L (say, less than 0.1L). By Newton's Second Law, Md 2x/dt2 = -Mg(x/L), or d 2x/dt2 + (g/L)x = 0. This is just the equation we were talking about, so M moves harmonically with angular frequency = (g/L) 1/2. The frequency depends on the strength of gravity, and on the length of the cord, but remarkably not on the mass M, nor on the amplitude of vibration. The isochronous and harmonic nature of the motion depend on the smallness of the motion. For large swings, the pendulum no longer moves with simple harmonic motion! Finite motion is a different and more difficult problem (but one easily soluble). The way to solve more difficult problems of this sort was shown by J. L. Lagrange (1736-1813; Mécanique Analytique, 1788). One begins by selecting any convenient parameters, called generalized coordinates, that define the state of the system. If only one such parameter q is necessary, we have a system with one degree of freedom. For the pendulum, q could be the angle 2/2.
that the line OM makes with the vertical. The velocity of M can then be expressed in terms of the rate of change of , as v = Ld/dt. The kinetic energy T of M is then M(Ld/dt)
Lagrange showed that the partial derivative of T with respect to the rate of change of the coordinate is a generalized momentum. In this case, p = ML2(d/dt), which is the angular momentum about O, but the result also holds in general. Lagrange now expresses the potential energy V in terms of the coordinate, which in this case is MgL cos . The negative of the derivative of the potential energy with respect to the coordinate is the generalized force. In this case, it is -MgL sin , which is the torque about O. Newton's Second Law then gives MLd 2/dt 2 + Mg sin = 0, or d 2/dt 2 + (g/L)sin = 0. This is not our previous equation of motion, because the sine appears instead of just the angle. We won't solve it, since we just wanted to show how to use generalized coordinates to get equations of motion. The momenta and forces that are associated with a generalized coordinate are appropriate to the type of coordinate. Since we chose an angle, we get angular momenta and torques in place of linear momenta and forces. In our example, T was independent of q, and V was independent of q' = dq/dt. Lagrange also showed how to handle more complex cases by defining the Lagrangian function L = T - V, and then writing the equation of motion as d(L/q')/dt = L/q (the 's represent partial differentiation). This is Lagrange's equation, and further details are to be found in books on advanced mechanics. There is one such equation for each generalized coordinate, in systems with more than one degree of freedom. It is a very powerful way to find equations of motion. With very special choices of the generalized coordinates, each of Lagrange's equations may contain only a single generalized coordinate, and so the system is easily solved. These special generalized coordinates are called normal coordinates. Thus far, the pendulum does not move unless it is given an initial disturbance, say by being held away from the equilibrium position and released, or by being given a knock when hanging still. Our equations imply that the pendulum will continue moving with the same amplitude from then on, but we know from experience that the amplitude slowly decreases, and the pendulum eventually comes to rest, due to air resistance and friction in the support. To make the pendulum move continuously, it must be driven or forced. Suppose we do this by moving the support from side to side a distance y = D cos 't. We suppose D