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Unformatted text preview: PHYS 302: Unit 2 Viewing Notes – Athabasca University PHYS 302: Vibrations and Waves – Unit 2 Viewing Notes The lecture starts by adding oscillations using the trigonometric identity cos cos 2 cos cos , 2 2 which is discussed in both the Reading Notes and the textbook. With x1=Acosω1t and x2=Acosω2t, x1 x2 2 A cos 1 2 2 cos 1 2 2 . The second term (oscillation) is the “slow” one, varying at the difference of the frequencies, while the first is the “fast” one, which is the average of the two frequencies being added. The average frequency is simply the sum of the frequencies divided by two. If the frequencies are close to each other in value, then this average value is close to either or both of the original frequencies. The slow term modulates the fast term and produces beats. The frequency of the beats is Δf : the frequency difference. If the two oscillations do not have the same amplitude, then the beats do not feature complete cancellation. ˆ The main terms of friction force are F fr C1v C2 v v , respectively, the viscous and pressure terms (14:15). 2 Here we will consider only the viscous term appropriate to low speeds, and unless you are interested, you do not need to explore the general case further. At this point we can realistically talk about horizontal springs, since friction, which we wish to consider, would normally be expected with a horizontal spring. Changing notation of C1 to b, we find the equation of motion m kx bx . Defining k/m=ω02 and b/m=γ, this can be written x 2 x 0 x 0 . We want to find the frequency of the actual oscillation ω, which intuition tells us should be x lower (longer period) with friction than for completely free oscillation. The 18 minutes starting at 21:00 follow the French text very closely, doing a solution using complex numbers 2 (reproduced here for convenience). In the complex plane z 0 z 0 , and a solution is sought in the form z z Ae j ( pt ) . Then, ( p 2 j p 02 ) z 0 . Independently, the real and imaginary parts must be 0, and p must itself be complex for this to happen in a reasonable way. Put p=n+js (25:00), with n and s real. We need p2, which can be calculated as any binomial term, recalling that j2=-1: p2=n2-s2+2jns. Substituting, 2 ( n 2 s 2 2njs j n s 0 ) z 0 . Since z is not zero, the part in brackets must be zero, both in its real and imaginary parts. The imaginary part is -2njs+jγn=0, or γ/2=s (27:30). The real part, making this substitution, is 2 2 2 2 2 n 2 4 2 0 0 , that is, n 2 0 4 . At this point (28:40) p is fully determined, and t z Ae j ( pt ) Ae j ( nt jst ) Ae st e j ( nt ) Ae 2 e j ( nt ) (29:50). The first term is an exponential decrease that multiplies the second, which is an oscillatory term. To remind us that n is a frequency, we replace it with ω, with 2 2 02 4 . obviously less than ω02. Note an error at (33:50) in that the real part should remove the complex exponential. The real part is really x Ae t 2 cos(t ) . Also, T=2π/ω (35:50) (another correction). T does not change with time; it is only the amplitude that changes. We now introduce the quality factor Q and rewrite 1 2 02 4Q 2 . One use of Q is shown since A( N ) Ae N Q 0 , , meaning that Q/π is the number of oscillations that must take place for the amplitude to decrease by a factor of e (recall e≈2.7). Note the comment about 18 minutes (41:30) is likely correct, since some error corrections were spliced into the video! Professor Lewin demonstrates the decay amplitude of vibration of a Styrofoam ball from 27 cm amplitude to 10 cm amplitude (factor e). In this case Q is about 35, with a decrease in amplitude by a factor e after about 10 oscillations. Break 1 PHYS 302: Unit 2 Viewing Notes – Athabasca University After the return from break, near 48:00, Lewin refers to “8.02,” which is the MIT Electricity and Magnetism course. This course is taken by the MIT students, but a comparable course is not a prerequisite for PHYS 302 through Athabasca University. All needed material from that subject area will be presented in the PHYS 302 course materials. An RLC circuit is one which has a resistor, capacitor, and inductor, along with a switch and battery. R stands for resistance, C for capacitance, and L for inductance. (Why L for inductance? —Not sure, but possibly to honour the physicist, Lenz.) A resistor “resists” the flow of electricity, a capacitor “stores” electric charge, and an inductor “stores” magnetic field, which arises from current flowing in the inductor. Recalling the energy approach to oscillation, we could expect that the C acts like potential energy, while the L, reflecting flow, acts like kinetic energy. By exchange between these two forms, oscillation can take place. In turn, R acts like damping. Lewin proceeds with his analysis by looking at the electrical current flowing in the circuit (49:00). Since an electricity course is not a prerequisite for this course, we will review some basic concepts about electricity. The current in a circuit is due to the flow of electric charge and is denoted I. Charge itself is denoted q. At any point in the circuit, the rate of flow of charge is the current, so I dq dt (49:26). Another concept introduced here is the electric field, which is a vector denoted E. The electric field directs the motion of charge in a circuit or in space. In a resistor, the electric field draws charge through resistive material. Think of drinking a milkshake through a straw. The harder you suck, the quicker the milkshake gets into your mouth. You cannot drink a milkshake through a thin straw well, because you cannot suck hard enough. The pressure you apply to pull milkshake through a straw is very similar to the electric field acting on charge. The more electric field (suction) for a given size of resistor (straw), the more current (milkshake) will flow. A bigger resistor is like a thinner straw, with more resistance to flow. In the actual wires of the circuit and in the inductor, the electric field is very small (well approximated by zero). In a capacitor, charges are stored. Unlike gravity, which always goes into its source mass, electric field originates on (diverges from) positive charges and goes in toward negative charges. For this reason, there is an electric field across a capacitor between the effective positive and negative charges on its plates. Similarly, the battery that powers the circuit supports an electric field due to the chemical reaction taking place inside it. One of the famous laws of physics is Faraday’s Law. In fact, much of our electrically-powered life depends on this law. Faraday’s Law states that charges are not the only source of electric fields. Electric fields are also generated by changes in magnetic fields. A generator, or the alternator in a car, moves wires in the vicinity of magnets so that changing magnetic fields are present in the wires. The changing magnetic field acts as a source for an electric field, although such an electric field develops less directly than it does with a charge. Faraday’s Law relates the combination of derivatives of the B electric field (its “curl”) to the time rate of change of the magnetic field. This is written as E . In this t “differential” form, the time change of the magnetic field at a given point along the wire relates to (i.e., forces) the electric field to have spatial derivatives. Due to these spatial derivatives, E cannot be zero at all points of the circuit (even if it is zero at some point(s)); if one travels along the wire, one must pass through a non-zero electric field. If one adds these electric fields from all points along the wire, one ends up with a potential difference along the wire. Adding up the fields incrementally is called taking a path integral. Conceptually, a path integral is similar to an ordinary integral. You should have covered ordinary integrals in your prerequisite calculus course. In the case of E dl d , the circle through the integral sign means that the integral is not evaluated between two limits, but dt rather, following a path in space. Here, that path is our circuit. The circuit has a magnetic field through its area, and the total of magnetic field multiplied by area (also an integral) is called the magnetic flux. Faraday’s Law says that if the flux inside a path (in this case our circuit) changes, then its rate of change will cause an electric field to build up around the perimeter. If we go through a circuit, we can add up the bits making up the path integral of the electric field. These bits introduce another familiar concept: an electric field over a distance causes a voltage difference. Stated this way, this is may be a chicken-and-egg problem (What came first, the chicken or the egg?). Suffice it to say that in a capacitor, if there is charge on each plate of q (positive on one plate, negative on the other), then there is a voltage across the capacitor of VC=q/C, where C is the capacitance (capacity to store charge) of the capacitor (50:50). Across a resistor, the voltage drops by the product IR. Again, this is a chicken-and-egg problem. Basically, a voltage across a resistor causes current to flow in such a way that V=IR. (This is known as Ohm’s Law). Summing the voltages to determine E dl d is analogous to Kirchhoff’s Law, which Lewin mentions disparagingly dt 2 PHYS 302: Unit 2 Viewing Notes – Athabasca University (51:45). Lewin also mentions Maxwell, since Faraday’s Law is one of the Maxwell Equations (which you do not study in this course). One thing Lewin does not mention is that the flux is almost all contained inside the wire coil (the inductor), and for an inductor, the flux φ=LI, where L is the value of the inductor. In other words, current causes a magnetic field, and the flux of that field is proportional to the inductance times the current. The inductance is determined by the geometric properties of the inductor. Most inductors are coils, since winding up wire allows for many turns around the same area to get a bigger inductance in a small space. Since each turn encloses more flux, more flux gets enclosed for the same current. Doing the sum of the voltages, we get d IR 0 VC V0 d dt LI L dI (51:46). With I dt dt dq dt , this can all be brought to an equation in q: 2 q q L q L dI R dq C L d 2q R dq C V0 . Dividing by L and using ‘dot’ derivatives we get q R q LC VL0 . If dt dt dt dt we put R/L=γ and 1/LC=ω02, this starts to look very familiar (53:15): q q 0 q 2 V0 L , which is very similar to the equation for SHM of a spring with damping. The resistance plays a role in the damping (heat is dissipated in the resistor), and the capacitance and inductance determine the natural frequency—not surprising if oscillation is due to transfer of energy between them. Unlike the oscillation of a spring, however, the right-hand side is not zero. When all the time derivatives go away, physical intuition shows that oscillation ends when the capacitor is fully charged. In this case, we can write the spring solution but add this end result qmax=V0C, to a solution similar to that for a spring. Then q q1e t 2 cos(t ) qmax (55:54). This is a decaying oscillation, but it ends up at qmax. q1, and α must be determined from initial conditions. Solving shows q1 qmax / cos and tan 2 (58:00). For high-Q systems (and many RLC circuits are high-Q), q1≈-qmax to a very good approximation, so the oscillation starts very near 0 (1:01:00; graph). The demo that follows drives an RLC circuit with a square wave. (1:09:00) What if γ2/4> ω02? Then correction, this gives n j ( x A1e 1 2 2 [ ( 4 0 ) 2 ]t 2 A2 e 2 4 2 2 2 02 4 or equivalently, n 2 02 4 , is negative. Noting the 2 0 ) , and putting this back into z Ae j ( pt ) gives 2 1 2 [ ( 4 0 ) 2 ]t 2 , which is a pure exponential decay, corresponding to overdamping. Initial conditions determine the constants (1:14:00). A final, special case called critical damping is when ω0=γ/2. Then x ( A Bt )e t 2 , again, basically a decay. Note: The oscillatory case is called underdamped. The lecture ends with a demo of a damped torsional pendulum. 3 ...
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