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Other Physical Systems Sect. 3.7 • Recall: We’ve used the mass-spring system only as a prototype of a system with linear oscillations! – Our results are valid (with proper re-interpretation of some of the parameters) for a large # of systems perturbed not far from equilibrium & thus which have a “restoring force” which is linear in the displacement from equilibrium. – The “Restoring Force” in a particular problem might or might not be a real physical force, depending on the system. – The math (2nd order, linear, time dependent differential equation) is the same for such systems. Of course, the physics might be different. • SOME of the Mechanical Systems to which the concepts learned in our harmonic oscillator study apply: – Pendula (as we’ve seen in examples) including the torsion pendulum. – Vibrating strings & membranes – Elastic vibrations of bars & plates – Such systems have natural (resonance) frequencies & overtones. These are treated in identical manner we have done. • Acoustic Systems to which the concepts learned in our harmonic oscillator study apply: – In this case, air molecules vibrate – Resonances depend on dimensions & shape of container. – Driving force: a tuning fork or vibrating string. • Atomic systems to which the concepts learned in our harmonic oscillator study apply: – Classical treatment as linear oscillators. – Light (high ω) falling on matter causes atoms to vibrate. When ω0 = an atomic resonant frequency, EM energy is absorbed & atoms/molecules vibrate with large amplitude. – Quantum Mechanics: Uses linear oscillator theory to explain light absorption, dispersion, & radiation. • Nuclear systems to which the concepts learned in our harmonic oscillator study apply: – Neutrons & protons vibrate in various collective motion. – Driven, damped oscillator is useful to describe this motion. • Electrical circuits: Major examples of non- mechanical systems for which linear oscillator concepts apply! – This case is so common, people often reverse analogies & talk about mechanical systems in terms of their “equivalent electrical circuit”. – Discussed in detail next! Electrical Oscillators Sect. 3.8 in the old (4th Edition) book! In 5th Edition only in Examples 3.4 & 3.5 • Consider a simple mechanical (harmonic) oscillator: A prototype is shown here: • Equation of motion (undamped case): m(d2x/dt2) + kx = 0 Solution: x(t) = A sin(ω0t - δ) Natural Frequency: (ω0)2 (k/m) LC Circuit • Consider a simple LC (electrical) circuit: A prototype is shown here: (L = inductor, C = capacitor) • Equation of motion for charge q (no damping or resistance R): L(d2q/dt2) + (q/C) = 0 (1) Math is identical to the undamped mechanical oscillator! A more familiar eqtn of motion (?) in terms of current: I = (dq/dt). Kirchhoff’s loop rule L(dI/dt) + (1/C)∫Idt = 0 (2) Solution to (1) or (2): q(t) = q0 sin(ω0t - δ) Natural Frequency: (ω0)2 1/(LC) • A comparison of the equations of motion of mechanical & electrical oscillators gives analogies: x q, m L, k C-1, (dx/dt) I • Consider (let δ = 0 for simplicity): q(t) = q0cos(ω0t) [q(t)]2 = q02 cos2(ω0t) and I(t) = (dq/dt) = -ω0q0sin(ω0t) [I(t)]2 = [ω0q0]2sin2(ω0t) = [q02/(LC)]sin2(ω0t) So: (½)L[I(t)]2 + (½)[q(t)]2/C = (½)[q02/C] (1) With the above analogies, (1) is mathematically analogous to the total energy for the mechanical oscillator! We found: (½)m[v(t)]2 + (½)k[x(t)]2 = (½)kA2 = Em (2) From circuit theory, total energy for an LC electrical circuit is Ee (½)[q02/C] (1) is also analogous physically to (2)! • Physics: The total Energy of an LC circuit (½)L[I(t)]2 + (½)[q(t)]2/C = (½)[q02/C] = Ee = const.! • Physical Interpretations: (½)LI2 Energy stored in the inductor Analogous to kinetic energy for the mechanical oscillator (½)C-1q2 Energy stored in the capacitor Analogous to potential energy for mechanical oscillator (½)[q02/C] = Ee Total energy in the circuit Analogous to the total mechanical energy E for the SHO Also, Ee = constant! The total energy of an LC circuit is conserved. The system is conservative! (Only if there is no resistance R!). As we’ll see, in electrical oscillators, R plays the role of the damping constant b (or β) for mechanical oscillators. Example 3.4 (5th Edition) • Consider a vertical mass-spring system: ~ Similar to a free oscillator, but there is the additional constant downward force of the weight F = mg. At equilibrium, the weight stretches the spring a distance h = (mg/k) There is a new equilibrium position at x = h The eqtn of motion is the same as before with x x - h . So, it is: m(d2x/dt2) +k(x-h) = 0 with initial conditions x(0) = h +A, v(0) = 0 Solution: x(t) = h + A cos(ω0t) • Analogous electrical oscillator system to the vertical mechanical oscillator? • LC circuit with a battery (a constant EMF source ε)! • Equation of Motion? Kirchhoff’s loop rule gives: L(dI/dt) + (1/C)∫I dt = ε = [q1/C] q1 Charge that must be applied to C to produce voltage ε • With I = (dq/dt) this becomes: L(d2q/dt2) + [q/C] = [q1/C] (1) • (1) is mathematically identical to the mass-spring system with a constant external force (gravity). For initial conditions: q(0) = q0, I(0) = 0, solution is: q(t) = q1 + (q0 - q1) cos(ω0t) • This circuit is an exact electrical analogue to the vertical spring-mass system in a gravitational field. LRC Circuit • Recall the mechanical oscillator with damping: • Equation of motion: m(d2x/dt2) + b(dx/dt) + kx = 0 • We’ve seen that the general solution is: x(t) = e-βt[A1 eαt + A2 e-αt] where α [β2 - ω02]½ A1 , A2 are determined by initial conditions: (x(0), v(0)). ω02 (k/m), β [b/(2m)] We’ve discussed in detail the Underdamped, Overdamped, & Critically Damped cases. • Analogous electrical oscillator system to the damped mechanical oscillator? • An LRC circuit is an electrical oscillator with damping. • Equation of Motion: Kirchhoff’s loop rule: L(dI/dt)+RI + (1/C)∫I dt = 0 (1) In terms of charge, I = (dq/dt), (1) becomes: L(d2q/dt2) +R(dq/dt) + (q/C) = 0 (2) (2) is identical mathematically to the damped oscillator equation of motion with x q, m L, b R, k (1/C) General Solution is clearly q(t) = e-βt[A1 eαt + A2 e-αt] with α [β2 - ω02]½ ω02 (LC)-1, β [R/(2L)] Could discuss Underdamped, Overdamped, & Critically Damped solutions! Summary of Electrical-Mechanical Analogies From the last row, clearly, the mechanical oscillator-electrical oscillator analogy also carries over to the driven mechanical oscillator driven circuit.We’ll briefly discuss this soon. Mechanical Analogies to Series & Parallel Circuits • We’ve just seen: – The mechanical oscillator with spring constant k is analogous to the inverse capacitance (1/C) = C-1 in an electrical oscillator. – Inversely, the mechanical compliance (1/k) = k-1 is analogous to the capacitance C • Consider a circuit with 2 capacitors C1, C2 in parallel – From circuit theory, the effective capacitance is Ceff = C1+ C2 • For 2 capacitors C1, C2 in parallel Effective capacitance: Ceff = C1+ C2 • Consider 2 springs with constants k1, k2 in series – Effective spring constant (effective compliance): (1/keff) = (1/k1)+ (1/k2) • Proof: Apply a force F to 2 springs in series: – Spring 1 will extend a distance x1 = (F/k1) spring 2 will extend a distance x2 = (F/k2). Total extension: x = x1+x2= F[(1/k1)+(1/k2)] (F/keff) 2 springs in series are analogous to 2 capacitors in parallel! • The mechanical oscillator with spring constant k is analogous to the inverse capacitance (1/C) = C-1 in an electrical oscillator. • Inversely, the mechanical compliance (1/k) = k-1 is analogous to the capacitance C • Consider a circuit with 2 capacitors C1, C2 in series – From circuit theory, the effective capacitance is (1/Ceff) = (1/C1) + (1/C2) • For 2 capacitors C1, C2 in series Effective capacitance: (Ceff)-1 = (C1 )-1 + (C2)-1 • Consider 2 springs with constants k1, k2 in parallel – Effective spring constant: keff = k1+ k2 • Proof: Stretch 2 springs in parallel a distance x: – Spring 1 will experience a force F1 = k1x, spring 2 will experience a force F2 = k2x. Total force: F = F1+F2= (k1+k2)x keff x 2 springs in parallel are analogous to 2 capacitors in series! AC Circuits • AC circuits (sinusoidal driving voltage E0sin(ωt)) are analogous to the driven, damped oscillator. – The mathematics is identical! – Can get resonance phenomena, etc. in exactly the same way as for the mechanical oscillator. – Can carry the mechanical oscillator results over directly using x q, m L, k C-1, v = (dx/dt) I = (dq/dt) (ω0)2 = (k/m) 1/(LC), β R F0sin(ωt) E0sin(ωt) – Results in both current & voltage resonances. See Example 3.5, 5th Edition, which does this in detail!

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