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Unformatted text preview: PHYS 302: Unit 4 Viewing Notes – Athabasca University PHYS 320: Vibrations and Waves – Unit 4 Viewing Notes Steady-state driven solutions have no adjustable constants. Initial conditions appear to be forgotten. After a long time, the driving totally dominates the situation, which explains why initial conditions play a small role. In the case considered, the undamped oscillation frequency would be ω0=k/m, and the damping constant γ=b/m. The undriven solution was x = X e-γt/2cos(ω’t+α), where X and α come from initial conditions. Here, ' 02 2 4 , which is slightly less than the original natural frequency. This is a decaying oscillation. With driving, if we apply force to m and drive at frequency ω with force F0cosωt, we will find x = Acos(ωt – δ), with tan 2 2 0 and A F0 / m ( 2 ) 2 ( ) 2 2 0 . These have no adjustable parameters, but of course due to the driving, the motion does not decay. Even though energy dissipates, the driving adds “fresh” energy. F0 cos t , where the top is for driving the m 2 0 0 cos t 2 2 Undriven, we had x 0 x 0 . Driven, x 0 x x x mass, and the bottom applies if the end of the spring is driven. The bottom form is not worked out but comes from F=kx when motion of ξ0 is applied (recalling that k/m=ω02). General solution: x = Acos(ωt – δ) + X e-γt/2cos(ω’t+α), with the second term being the transient, and the first the steady state solution. We could, in principle, solve for the constants, but this would be algebraically challenging. Problem 2.5 addresses this. It is possible to get the transient and steady state beating, especially if Q is high. dW F dx , and power P=dW/dt, so F v Fv F0 cos t[sin(t )] F0 cos t[sin t cos cos t sin ] . Usually only the average Work power is of interest. Time averaging, the cos*sin term is 0; cos*cos term averages to ½. Average power P 1 F0A sin 2 F02 2 2 2m 0 2 2 . We can examine this equation. If damping is very large, the average power is 0. This comes out of the large γ in the denominator. If m is very large, no force can get the mass going, and the power again goes to 0. If F0 is zero, nothing moves, so the power also goes to 0. If drive frequency is 0, again nothing moves, and power again equals 0. If drive frequency is very high, inertia prevents much motion, and again power goes to 0. If drive frequency approaches the natural frequency, maximum power is used: Pmax F02 QF02 , since Q=ω0/γ. The curve of power input resembles a resonance curve, and it can be 2m 2m 0 shown that half-width (in terms of frequency difference) is very close in value to γ. A highly damped system has a wide resonance curve. Break (40:00) 1 PHYS 302: Unit 4 Viewing Notes – Athabasca University For an RLC circuit (discussed in MIT 8.02 Electricity course, where it was driven with a battery), we now consider an oscillatory voltage source. Background information on RLC circuits is found in the Reading Notes for this unit and in the textbook. In the circuit, charge q is on the capacitor plate and the current is I=dq/dt; at the capacitor V0=q/C. Faraday’s Law applies, so d inside the whole circuit where φ is the magnetic flux enclosed; most of this is enclosed in the E dl dt inductor L by design. Applying Faraday’s Law around the whole circuit, IR+0+VC-V0cosωt=-L dI/dt. Clearly φ=LI; this defines the inductance. Taking another time derivative, LI RI V0 sin t I 2 V0 sin t , or I I 0 I L C (some definitions crept in—γ=R/L—but this is amazingly similar to the spring equation). Reactance (units = ohms) for the capacitor Z=1/ωC and Z=ωL for the inductor. There is similar resonant behaviour to that of a spring, with the reactances acting like damping. A plot of current flowing in the circuit as a function of frequency shows a resonant peak of current of V0/R (amount) at the frequency ω0. Atomic physics: Electrons have discrete orbits with discrete energy levels, and they can make transitions with changes of energy, which are carried into or out from the atom by electromagnetic radiation. The energy of light is proportional to its frequency (red is low energy; violet is high energy). The lower atmosphere of the Sun radiates all frequencies as a continuous spectrum (black body)*. As light having this spectrum exits through the atmosphere, resonant absorption by gas in the atmosphere removes certain frequencies. These are observed as “spectral lines,” because less light comes out at those frequencies. Spectral lines allow identification of the elements present in the Sun (and other stars). Helium (Greek for Sun is Helios) was first found on the Sun, using one of its characteristic spectral lines; helium was previously unknown, having never been observed on Earth. Regarding the demonstration on gratings, gratings are not covered at this stage in this course, but were covered in Athabasca University’s PHYS 202. In the video lecture, very high-Q resonant absorption on the atomic scale (of light by sodium atoms) is demonstrated using a grating. Note that the line appears dark because the absorbed light is re-emitted in all directions, so little of it goes back in the direction from which it was absorbed (i.e., back towards the direction of the incident beam of light). At this point in the course, regard a grating as a way to generate a spectrum of light. Near the end of the course we will see how this ability to spread out the colours of light actually arises from the wave nature of light. *Spectra, including black body spectra, will be addressed in the readings and videos on Quantum Mechanics, starting in Unit 8. At this point we stress that there are strong resemblances between resonant behaviour in the systems we have looked at in detail, and atoms. 2 ...
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