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We divide bymand letω0pk/mto rewrite the equation asx00+ω20x0.The general solution to this equation isx(t)Acos(ω0t)+Bsin(ω0t).By a trigonometric identityAcos(ω0t)+Bsin(ω0t)Ccos(ω0t-γ),for two different constantsCandγ. It is not hard to compute thatC√A2+B2andtanγB/A. Therefore, we letCandγbe our arbitrary constants and writex(t)Ccos(ω0t-γ).Exercise2.4.1:Justify the identityAcos(ω0t)+Bsin(ω0t)Ccos(ω0t-γ)and verify theequations forCandγ. HintStart withcos(α-β)cos(α)cos(β)+sin(α)sin(β)and multiplybyC. Then what shouldαandβbeWhile it is generally easier to use the first form withAandBto solve for the initialconditions, the second form is much more natural.The constantsCandγhave nicephysical interpretation. Write the solution asx(t)Ccos(ω0t-γ).This is a pure-frequency oscillation (a sine wave). TheamplitudeisC,ω0is the (angular)frequency, andγis the so-calledphase shift. The phase shift just shifts the graph left or right.We callω0thenatural (angular) frequency. This entire setup is calledsimple harmonic motion.Let us pause to explain the wordangularbefore the wordfrequency. The units ofω0are radians per unit time, not cycles per unit time as is the usual measure of frequency.Because one cycle is2πradians, the usual frequency is given byω02π. It is simply a matter ofwhere we put the constant2π, and that is a matter of taste.Theperiodof the motion is one over the frequency (in cycles per unit time) and hence2πω0. That is the amount of time it takes to complete one full cycle.Example 2.4.1:Suppose thatm2kgandk8N/m. The whole mass and spring setupis sitting on a truck that was traveling at 1m/s. The truck crashes and hence stops. Themass was held in place 0.5 meters forward from the rest position.During the crashthe mass gets loose. That is, the mass is now moving forward at 1m/s, while the otherend of the spring is held in place.The mass therefore starts oscillating.What is the
. . MECHANICAL VIBRATIONS107frequency of the resulting oscillation? What is the amplitude? The units are the mks units(meters-kilograms-seconds).The setup means that the mass was at half a meter in the positive direction during thecrash and relative to the wall the spring is mounted to, the mass was moving forward (inthe positive direction) at 1m/s. This gives us the initial conditions.So the equation with initial conditions is2x00+8x0,x(0)0.5,x0(0)1.We directly computeω0pk/m√42. Hence the angular frequency is 2. The usualfrequency in Hertz (cycles per second) is2/2π1/π≈0.318.The general solution isx(t)Acos(2t)+Bsin(2t).Lettingx(0)0.5meansA0.5. Thenx0(t)-2(0.5)sin(2t)+2Bcos(2t). Lettingx0(0)1we getB0.5. Therefore, the amplitude isC√A2+B2√0.25+0.25√0.5≈0.707.The solution isx(t)0.5 cos(2t)+0.5 sin(2t).