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Other Physical Systems Sect. 3.7

Other Physical Systems Sect. 3.7. Recall: We’ve used the mass-spring system only as a prototype of a system with linear oscillations !

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Other Physical Systems Sect. 3.7

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  1. Other Physical SystemsSect. 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.

  2. SOME of the Mechanical Systemsto 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 Systemsto 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.

  3. Atomic systemsto 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 = anatomic 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.

  4. 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!

  5. Electrical OscillatorsSect. 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)

  6. LC Circuit • Consider a simple LC(electrical)circuit: A prototypeis 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)

  7. 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 isEe (½)[q02/C]  (1) is also analogous physically to (2)!

  8. 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.

  9. 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 thesame 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)

  10. 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.

  11. 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.

  12. Analogous electrical oscillator systemto 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]½ω02 (LC)-1, β  [R/(2L)] Could discuss Underdamped, Overdamped, & Critically Damped solutions!

  13. Summary of Electrical-Mechanical Analogies From the last row, clearly,the mechanical oscillator-electrical oscillator analogy also carries over tothe driven mechanical oscillator  driven circuit.We’ll briefly discuss this soon.

  14. Mechanical Analogies to Series & Parallel Circuits • We’ve just seen: • The mechanical oscillator with spring constant k is analogousto the inverse capacitance (1/C) = C-1 in an electrical oscillator. • Inversely, the mechanical compliance  (1/k) = k-1 is analogousto the capacitance C • Consider a circuit with 2 capacitors C1, C2in parallel  • From circuit theory, the effective capacitance is Ceff = C1+ C2

  15. For 2 capacitors C1, C2in 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!

  16. The mechanical oscillator with spring constant k is analogousto the inverse capacitance (1/C) = C-1 in an electrical oscillator. • Inversely, the mechanical compliance  (1/k) = k-1 is analogousto the capacitance C • Consider a circuit with 2 capacitors C1, C2in series  • From circuit theory, the effective capacitance is (1/Ceff) = (1/C1) + (1/C2)

  17. For 2 capacitors C1, C2in 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!

  18. 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 usingx  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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