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« PHYS521 Fall 2015
Homework 12 Pointers (Last Homework!)
Sat 05Dec2015 11:47PMThis final set deals mostly with time-dependent E&M fields (including radiation). The last question is on the Lorentz transformation of the fields and, though that is not an area of heavy emphasis in the recent Qualifying Exams, remains to be a very important and interesting part of fundamental E&M. It is also something you would be at a loss to deal with if a problem appeared and you haven’t thought about it or encountered it at all.
Problem 1: Note that the xy-plane is vertical in the given diagram. This problem is ill-defined unless the radii of the “tiny spheres” are given; why? Let’s say the radius is r. You can then proceed to assume that charges q(t) and –q(t) are on the tiny spheres – they don’t enter the final answer.
Problem 3: Can you write down the dimensions of m (magnetic moment) and fundamental constants like μ0 and c?
Problem 4: You need to be able to handle lots of vector cross products.
Problem 6: If you use a complex representation for E and B, i.e., E=E0ei(k·r-ωt), B=B0ei(k·r-ωt), then E0 and B0 are generally complex, mutually perpendicular vectors. For dissipating medium like a conductor, we must allow k to have an imaginary part (say, k=k0+iκ). Then use Maxwell's equations to relate the phase between B and E to that of k, and solve for that.
Problem 8: Can you express the current density J using a θ-function? Then the rest is doing an integral.
Problem 9: The power flux per unit solid angle from a dipole antenna in the far-field region is given by dP/dΩ ~ (proportional to) k4 |p|2 sin2θ where p is the dipole moment and θ is the angle the line of sight from the antenna to the observation point makes with the vertical.
Problem 10: In the actual Qualifier, the equation sheet contained the Lorentz transformation of the E and B fields. For part (b), you should note the relationship of your computed answers and your expectations based on what should be a Lorentz invariant.
Homework 11 Pointers (Early!)
Thu 19Nov2015 5:05PMFirst, Happy Thanksgiving to you all! Since you have plenty of time to work this set (till 12/2), You may wish to hold off on reading the following comments/hints and try them by yourself first. Then come back here for some hints.
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This set is mostly a collection of problems that involve a current loop (or a magnetic dipole) with a couple of problems involving electric dipoles and Stat Mech thrown in.
Problem 1: While you could work this problem with brute force as an application of the Biot-Savart's law without much difficulty, you can do it a bit, or a lot more easily using some, or much, symmetry and knowledge from related geometry. In the latter case, you either don't need the second integral, or neither of the integrals provided. If you are interested in saving efforts, think about what simple device this problem reminds you of.
Problem 2: This does require integrals of the Biot-Savart’s law. There seems to be no way around it. Still, using suitable change of variables and symmetry, etc., does help you a lot.
Problem 3: For (a), what are the boundary conditions that B must satisfy at z=0? For (b), Don’t forget that the energy U provided is for two real dipoles, whereas in this problem, one is real and the other is a fictitious image dipole. In (c), what is the relationship between the surface current density and the magnetic fields on both sides of the surface?
Problem 6: You can either use the form of Biot-Savart’s law given or one that’s more pedestrian.
Problem 7: The electric quadrupole contribution to the electrostatic potential is equal to (1/4πε0)Q/r3 where Q is supposed to be given in the problem’s Hint. The only problem is, the Hint is incorrect as it stands and we should replace ri by r’i and θi by θ’i where r’i is the distance from the origin to the source i and θ’i is the angle made by the segment from O to source i and the segment from O to the observation point A. The way it stands, the ri given in the figure do not correspond to those variables in the Hint! This form of Q is derived, e.g., in Griffiths, Chapter 3. This Q is a bit different from what is usually called a quadrupole tensor (it’s a 3x3 matrix rather than a number) (see, e.g., Jackson), but corresponds to the same term in the multipole expansion of the electrostatic potential.
Problem 8: First consider a single (electric) dipole and focus on the angular average. (The positional and momentum degrees of freedom are irrelevant in this problem.) Then you can define an orientational partition function and the desired average from it. Finally, you use the given concentration to arrive at the polarization.
Problem 9: The constraint given on the normal zipper operation tells you what the allowed states of the system are.
Problem 10: Interface (or surface) tension coefficient here is the excess interface (or surface) free energy per unit area when the interface is present. The relationship given for the Δp, R, and σ follows from what’s called Young-Laplace equation. It states that Δp = - σ div(n) where n is the unit normal pointing out of the surface.
No PHYS521 class next Monday, Nov. 23.
Wed 18Nov2015 4:43PMNo PHYS521 class next Monday, Nov. 23 as we already compensated for it by having a couple of extended classes. Since there is no class for Purdue generally on Wed., Nov. 25, PHYS521 will have no class next week. Homework 11 (current set) will be due on Wednesday, Dec. 2. I will post some hints early because I will be travelling, but you may choose to hold off on looking at them until you gave the set a genuine attempt yourself first.
Happy Thanksgiving!
Hisao Nakanishi
Homework 10 Pointers
Sat 14Nov2015 3:20PMThe E&M problems in this set are either on induction (Faraday’s Law) or on superconductors. The last two problems are somewhat complicated examples from statistical mechanics where you need to calculate the entropy of a system that’s neither a simple ideal gas nor a harmonic oscillator.
Problem 1: This is another problem where the “Factor of 2” is relevant. An external source must keep the constant current (free current), doing work against induced EMF. What is constant through the inside of the solenoid?
Problem 3: The physics here is that the momentum and angular momentum in the E and B fields initially are transferred to the mechanical momentum and angular momentum of the charges as time goes on. E and B are related by Faraday's law.
Problem 4: Get a differential equation for (d/dt)θ.
Problem 7: Consider Meissner effect where B=0 except on the surface of a superconductor (or, actually, close to the surface – see Problem 8).
Problem 8: Ordinarily, we consider B=0 inside a superconducting slab. However, in this problem, we are considering the evanescent penetration of B into the slab. Use known boundary conditions for B.
Problem 9: (a) Estimate how many atoms are in the excited state with internal energy Δ vs 0 to calculate the mean energy at T2. (b) Total entropy is the sum of the internal entropy due to the two available states and the translational entropy.
Homework 9 Pointers
Sat 07Nov2015 1:48PMThe first 5 E&M problems in this set deal with forces acting on charges and currents due to electromagnetic fields and Problems 6-8 have to do with time-varying currents. The two statistical mechanics problems ask you to calculate entropy for systems with discrete degrees of freedom such as spins or two level particles.
Problem 1: Assume the tension and curvature to be constant throughout the wire (by symmetry). For function f(x) in general, the radius of curvature is R = (1+f’2)3/2/f’’ (why?).
Problem 4: For part (b), note T = ∫ from y0 to 0 (dt/dy) dy, a common trick for this type of a problem.
Problem 5: Note that there should not be a minus sign in the formula given for E of the radiation due the accelerating charge. Think very carefully about the time and length scales in the problem. The interpretation of “just after time t3” I suggest for you to adopt is t3+ta where ta is the duration of the acceleration of the charge (O(10-12 sec)).
Problem 6: This is just the series LRC circuit less the inductor L.
Problem 7: First show the sign convention explicitly in a diagram. The given equations make sense only with a consistent sign convention (as to the positive directions of the currents and those of the induced potentials). The voltage on each side of the transformer is the combination of contributions from self-induction and mutual induction.
Problem 8: Draw and use an equivalent circuit diagram. It’s much easier to think of resistors and capacitors in such a diagram than to try to visualize the actual geometry.
Problem 10: Recall the statistical definition of temperature. Part (c) is essentially a derivation of the Boltzmann factor (not the use of it).
Homework 8 Some Answers
Wed 04Nov2015 5:34PMAs we ran out of time today and I could not tell you the answers and some steps for Problems 3, 5, and 6, I will describe them below.
Problem 3: Use method of superposition. The answer is a uniform magnetic field inside the hole: B = (μ0 J d)/2 in x-direction (perpendicular to the plane that contains the axes of the cylinder and the hole. To see this, you note that the vector (r eφ-r' eφ') is just (r eρ-r' eρ') rotated by π/2 counterclockwise.
Problem 5: The momentum density is non-zero only within the cylinder. For a point of observation (ρ,θ,z) in the cylindrical coordinates relative to the cylinder axis, the total angular momentum comes down to the integral over the cylinder of the quantity Qμ0I/(4π2ρ'L) cos(θ' - θ) eθ, where the primed quantities refer to the cylindrical coordinates relative to the current axis. The difficult part is to convert this cos(θ' - θ)/ρ'. Use the law of cosines to the triangle made by sides ρ, ρ', and 2R to convert this into (ρ-2R cosθ)/(ρ2-4Rρcosθ+4R2). Then note that the x-component of eθ vanishes after the integration. The final answer is: L = -(Qμ0I)/(16π) ey.
Problem 6: This problem is also difficult because the proper integration requires going back and forth among different coordinate systems. You can start with a mixture of spherical and Cartesian coordinates initially, but the Cartesian part quickly becomes the unit vector in z-direction and the scalar part becomes a spherical integration that contains the integral that is given in the problem. The end result is L = ez (μ0 qe qm)/(4π). Remarkably, this is independent of d. It also implies that, if L is quantized (discrete), then the charges (electric and magnetic) must also be quantized.
Homework 8 Pointers
Sat 31Oct2015 10:55AMMost of the E&M problems in Homework 8 deal with a constant current situation (as opposed to static charges in Homework 7), but Problem 6 is one where a static electric and a hypothetical static magnetic charge are present. This problem is included in this set because it asks you to calculate the angular momentum of the electromagnetic fields, similarly to Problems 4 and 5. The last 3 problems are from Stat Mech and test your understanding of the probability distribution (partition function, Boltzmann factor, entropy, etc.).
Problem 1: The power transported by the cable is the integral of the Poynting vector over the cross sectional area.
Problem 2: It's a problem of Ampère's law, but in which form is it most useful here? Then, what is continuous across the "small gap"?
Problem 3: Principle of superposition. Then, use geometry to simplify the final B (both direction and magnitude).
Problem 4: The angular momentum is L = r x p where the general expression for the linear momentum density for the electric and magnetic fields is given in the problem. Note both must be present for p to be non-zero.
Problem 5: This is an integral problem as you can see from the two integrals provided. As such, choosing the right coordinate system that takes advantage of the symmetry and simplifying the integral at each step will be helpful. You do need patience and care to detail to get this right.
Problem 6: It's simplest to first simplify the integrand for L, using both spherical and Cartesian coordinates as well as symmetry. A lot of terms drop out, and the integral should reduce to the one provided. The answer will be independent of d!
Problem 7: You must use Ohm’s Law and the relationships between the conductivity σ and resistance R, but otherwise this is a Gauss’s Law problem.
Problem 8: In (b), “adiabatically” means keeping the entropy constant.
Problem 9: Probability density P(x) that the particle location is x can be obtained from the Boltzmann factor by integrating all other degrees of freedom (and normalizing by the partition function, which integrates all degrees of freedom including x).
Problem 10: It's for a classical particle.
Solution to Homework 7, Problem 3 posted under Reference Materials.
Wed 28Oct2015 4:44PMAs in the title ...
Homework 7 Pointers
Sat 24Oct2015 10:41AMHere are some pointers for Homework 7. Most of the E&M problems here are related to Gauss's Law. The two stat mech problems are from classical statistical mechanics.
Problem 1: Gauss's Law in the two forms (one for E and the other for D) and the relationship between the two give you all you want to know.
Problem 2: A number of ways to do this. If you wish to do it in the order of the question parts (i.e., (a) first, then (b), etc.), then you can calculate the overall capacitance by modelling the capacitor as the series combination of thin slices of capacitors (with varying capacitance). Then use the two forms of Gauss's Law. You could also do (b) first, then (a), (c).
Problem 3: It's probably best to absorb c0 and cn in the Hints into F0 and Fn. Then Laplace equation applied to the Fourier series allows you to solve for F0 and Fn. Here, don't forget that Laplace equation is valid only in upper and lower half spaces and NOT on the xy-plane. So the solutions for F0 and Fn in general involve different coefficients for the two half spaces. Then use symmetry to settle the differences if any.
Problem 4: It is simplest to consider the metal ball and plate to have a capacitance Cb and Cp, respectively. This way, you can relate the potential and charge in a simple way.
Problem 6: The “Factor of 2” issue is relevant here. You have to include the work being done by the external power source just as we discussed in class for the parallel-plate capacitor problems (when the plate separation is changed or when dielectric media is made available to the space between the plates). Otherwise you may get an answer with correct magnitude which, however, makes no physical sense.
Problem 7: Don't forget that the pressure acting on a surface charge density σ is σ2/2ε0 (factor of 2 there!). Do the integral carefully (taking into account the directions of the force).
Problem 8: You can approach this directly by using forces or by using the potential energy. If you do the latter, this is another problem where the external source (battery) cannot be excluded from consideration.
Problem 9: Entropy of mixing. We will review the classical ideal gas in class, but in short, the partition function of an N-particle, monatomic, classical ideal gas is ZN = z1N/N! where z1 = nQ V and nQ = [(mkBT)/(2π hbar2)]3/2. FN= –kBT ln ZN and S=−(∂F/∂T)|T.
New Grader
Thu 22Oct2015 9:48AMWe have a new grader for the second half of the course: Mr. Xiaobing Shi. He will hold his office hours at the same time slot as Mr. Chen did, i.e., Mondays, 4:30 - 5:30 pm, but in Rm.105.
Midterm Exam and Extended Class(es)
Thu 08Oct2015 10:03AMOur midterm exam will be given 2:30 - 4:30 pm on Monday, Oct. 19 in ARMS 1103. Please note the change in location - it is in Armstrong. It will have 5 actual past Qualifying Exam questions from the topics covered in class or closely related to those up to that time.
Since we are using the class hour for the exam and yet cancelling one class (on Nov. 23), I will have extended classes on Wed., Oct. 21 and Monday, Oct. 26. These classes will begin at 2:30 (instead of 3:00) in the regular classroom. I hope all of you can make them.
Thank you.
Hisao Nakanishi
Homework 6 Pointers
Sun 04Oct2015 12:11PMHW#6 collects one-of-a-kind problems as well as those from relatively infrequently posed ones (at least in recent years) including rotating frame of reference and relativity. For basic relativistic formulae, please refer to a short excerpt of the notes for Special Relativity that I posted under Reference Materials section. The notes come from my class on mechanics at the junior/senior level and are based on the text by Thornton and Marion.
Problem 1: If you cut a spring of spring constant k and length L in half, what is the spring constant of each half?
Problem 2: Assume that the student loses contact with the floor when s/he is stretched to full height. Think about the energetics.
Problem 3: K (he kinetic energy) = E (total energy) - U. Classically, a particle cannot be at a position where U>E since K>=0. A phase diagram (in x-p space) is a set of trajectories and as such you should not forget to draw arrows to indicate the directions of flow as well.
Problem 4: Likely the hardest (and an oddball) problem even of this lot. It may simplify algebra to set up a coordinate system translated by R in both horizontal and vertical directions from the floor/wall. In principle, this is a simple problem of energetics (with a geometric constraint). A key is to express the velocity of the bottom cylinder solely in terms of one trigonometric function of an angle.
Problem 5: Think about time dilation and relativistic Doppler effect. Also, what is the relationship between the total number of pulses Ulysses emits and that of the pules Homer receives?
Problem 6: Do you remember the conservation laws in relativistic formulation?
Problem 7: For the net force in an inertial (lab) frame in polar coordinates, you can refer to the previous post under the Reference Materials.
Problem 8: Use equipartition theorem for (a). For (b), use an argument (without any calculation). Think about how quatum and classical calculations of things like heat capacity are related.
Problem 9: You can assume that the entropy S_s in the superconducting phase goes to zero as T→0. Also, since the entropy S_n in the normal phase would asymptotically be linear in T as T→0 IF there were no intervening superconducting phase transition, you can also assume that S_n goes to zero as T→0.
Problem 10: The formula given in the problem for the chemical potential μ_s of the solvent molecule in a solution comes from the (extensive) Gibbs free energy of a dilute solution (cf. Landau and Lifshitz in the chapter for solutions, e.g.):
G(T,P,N_s,N_b) = N_s μ_0 + N_b k_B T ln[(N_b/e) (f(T,P)/N_s)]
where μ_0 is the solvent chemical potential in a pure solvent (not a solution).
Thus, μ_s = (∂G/∂N_s)|_(constant N_b,T,P) = μ_0 - (N_b/N_s) k_B T as given in the problem. Anyway, you can just use this relation in this problem, not derive it.
In this problem, use the fact that, in equilibrium, the solvent chemical potential in the pure solution (on the right) and that in the solution with solute molecules also present (on the left) must be equal. Since the solute cannot pass through the membrane, the solute chemical potential doesn't have to match; it's the solvent chemical potential that must match between the right and left halves.
Then, interpret that relation in terms of a Taylor series expansion of the solvent chemical potential in pressure and use the second relation that is also given in the problem.
Mathematically, this (and similar) problem is very simple, but if you are not used to this type of a problem, it may bewilder you as to the significance and origin of terms, relations, and results.
Homework 6 Due Date Extended
Thu 01Oct2015 12:31PMHomework 6 will now be due on Wednesday, Oct. 14, due to a number of students having exams in other courses during the week of Oct.5.
Homework 5 Pointers
Sun 27Sep2015 11:25AMHomework #5 collects problems that involve friction or other drag force from mechanics and two additional problems from thermal physics. The latter problems involve qualitative thermodynamic arguments and some quantitative estimation.
Problem 1: Though no word "friction" appears in the problem, it does say "rough horizontal table". This implies that you should use (standard) kinetic friction that is proportional to the normal force. Some force must cancel the tension in the string in order for the system to have constant velocity, and it would have to be the friction.
Problem 2: There are many possible ways the three blocks can move with respect to each other and to the surface, but which one actually happens? Use various arguments/equations to rule out unphysical assumptions. As an example, if you suppose all 3 blocks move together (i.e., the two upper blocks do not slip on the bottom one), then you can show that the whole piece (3 blocks) must accelerate opposite to the pulling force, which is a contradiction.
Problem 3: Assume that the static friction coefficient is greater than the kinetic one, which is a reasonable assumption.
Problem 4: For part (B), what is the physical condition that the roller coaster is still on the rails at the top of the loop (upside down)?
Problem 6: Are both (1) and (2) stable?
Problem 7: Stokes drag on a spherical body is proportional to the velocity and radius with the proportionality constant equal to 6πη.
Problem 8: As in more complex collision problems, think about the process in multiple stages, i.e., the initial state, the duration of the "collision" where A is sliding over B and C, and the final state where A is exactly on top of C.
Problem 9: Just draw an isotherm, starting from where all of the material is a gas at a low density/pressure, and decrease volume (increase pressure) at constant temperature. At some point, you enter a two-phase region where part of it starts to become liquid. Eventually, the density (volume) is high (low) enough that all material becomes liquid. While in the two-phase region, the pressure remains constant; otherwise, pressure increases when volume is decreased. This is the analog of the well-known situation when you cool liquid at constant pressure and it starts to freeze. While the freezing is going on, the temperature remains constant, but it starts to decrease again after all material has become solid.
Problem 10: The particle flux onto the hole can be related to the pressure and momentum change of a particle if it hits the wall and were to be reflected back. This can be calculated exactly if the velocity distribution is known, but in this problem, you are only asked to estimate. So the Equipartition Theorem comes in handy.
Homework 4 Pointers
Sat 19Sep2015 9:02AMHomework #4 collects problems of central force motion and those of heat cycles (engines and refrigerators) for the most part. Inverse-square central forces should already be familiar to you (if not, please review your mechanics text for them, e.g., Chap.8 in both Thornton and Marion and Taylor, or Chap.7 in Morin). They end up in Kepler’s laws and conic section trajectories – circle, ellipse, parabola, or hyperbola, which we will briefly review in class on Monday as well. Heat cycles use ideas of the first and second laws of thermodynamics liberally; if not familiar, please review, e.g., Chap.4 of Schroeder, Chap.8 of Kittel and Kroemer, or Chap.6 of Zemansky and Dittman. In a historical introduction of classical thermodynamics, the Carnot cycle plays a fundamental theoretical role in establishing the Kelvin temperature scale, etc., but there are also quite practical applications of heat cycles.
Problem 2: The stability analysis in part (b) is similar to what you already encountered in Homework 3.
Problem 3: Draw pictures! There are a number of facts and identities that you need to use in this problem (rather than re-deriving everything again!), which we will discuss in class on Monday.
Problem 4: This problem is a bit non-standard way of studying essentially the same Kepler problem as you normally do in a mechanics course. Do you see that it must lead to the same conic section?
Parts (a) and (b) are straight application of the definition of Hamiltonian and Hamilton’s equations. We did derive them in our first class of the semester, but you may wish to review them again in your favorite mechanics text. Part (c) is easiest if you just think about the torque/angular momentum relationship. Parts (d) and (e) may be hard! My suggestion is to take advantage of the result of (c) and orient the spherical coordinate system in such a way that the plane containing the motion is a nice simple plane (e.g., θ=π/2 or φ=0 or π) without loss of generality. Then split the possible cases by whether there is a finite range of allowed r or r can extend to infinity. The figure required is an example of a phase portrait.
Problem 5: Lagrange points are not exactly a part of household knowledge. There are 5 such points and the problem discusses one of them. Here, even though m is much smaller than M1 and M2, neither of the latter is infinite. So the center-o-mass of the latter two masses is the stationary point, and all 3 bodies orbit about this point with the same frequency.
Problem 8: Carnot refrigerator is a Carnot engine operated in reverse.
Problem 9: The result of (c) is identical to what is known as Clausius-Clapeyron equation. Usually, this relation is derived by equating the change in the Gibbs free energy per particle along the phase coexistence curve for just inside one phase (say A) and just inside the other phase (B), i.e., dμA = dμB. The current problem is cute in that it gives you the same result, but do note that it makes some bold assumptions (not justified) whereas the usual derivation is exact.
Problem 10: In part (b), assume that the 100W bulb left on inside the refrigerator is the dominant part of heat that needs to be removed to keep the inside of the refrigerator at constant temperature. (In other words, ignore heat coming in from outside the refrigerator into it, which may or may not be a reasonable assumption.)
About the factor of 3 in R vs y for Problem 3, Homework 3
Wed 16Sep2015 5:19PMWe (some of you and myself) had a brief discussion just after class about how to relate R and y in Problem 3 part (a) in Homework 3. After a little more thought, I believe that I have the reason why R ~ (A/4 rho) y (asymptotically) [without the extra factor of 3] is correct and the other answer with that factor of 3 is not. This is for the same reason that the volume of the cone is 1/3 that of the cylinder in which the cone is inscribed. Basically, you have to read the statement in the problem "The amount of the water deposited on the particle is proportional to the cross-sectional area of the particle times the distance traveled" as the statement on the rate (i.e., during a very small time duration). Otherwise, you will have to calculate the volume covered by the ever increasing particle cross-section (thus the cone), rather than multiplying the cross-section of the particle at one end of the time period (beginning or end) by the distance traveled. Those of you who insisted on the "3", think about this! Thanks.
H. Nakanishi
Correction on Problem Fall 2015 Part I 8
Tue 15Sep2015 1:47PMYesterday in class, I discussed the centrifuge problem (Fall 2015, Part I-8). In that discussion, I had completely forgotten about the gravity! So the answer I gave was incorrect, and I apologize for this error. There is both a contribution I discussed (coming from the distance from the axis, or z sin θ) and the contribution coming from gravity (coming from the vertical distance from the top, or z cos θ). So the correct answer has both terms in it. Please rethink about the problem yourself, and we can talk about it again if time permits.
Homework 3 Pointers
Sat 12Sep2015 8:47AMFirst, Mr. Jian-yu Chen's office hour has moved to Mondays, 4:30 - 5:30 pm (Rm.6A) as the previous office hour was conflicting with the 2 quantum mechanics courses that some of you are taking. Please take advantage of this opportunity!
Now, the Homework 3 pointers – beware there are some spoilers below! This set includes problems of several different kinds. First, I collected the problems in which the mass distribution changes in time. Then, there are problems in thermodynamics involving the black-body radiation and entropy changes of classical ideal gas. In addition, I have a couple of problems here that don’t belong to those types. The first type of problems are typically quite hard, that is, unless you approach it in certain ways. For black-body radiation, we will defer the derivation of the central result that J = σT4 where J is the energy flux density and σ is called the Stefan-Boltzmann constant. Rather, we just use it intelligently here. Likewise, we defer the statistical mechanical derivation of the classical ideal gas entropy and use a thermodynamic “derivation” of it instead. There, we can use some of the thermodynamic identities we already discussed in relation to ideal gas expansions previously. The remaining two problems, though rather different, the crux boils down to similar ideas of stability analysis. There, you perturb a stationary solution by a bit of variation and look at the equation of motion for the variation.
Problem 1: There are several points in time that delineates regions of different behaviors in this problem. What are they? The scale shows the normal force on the hour glass/sand, which balances the force acting on the hour glass/sand due to gravity and impulse by the falling sand. What is the latter equal to?
Problem 2: The “center of mass” in the problem actually refers to the midpoint of the rope.
Problem 3: Don't forget that the net force is equal to the rate of change of mementum (F=dp/dt). This is a hard problem.
Problem 4: Assume that the initial speed of the combined mass (m1+m2) as it takes off from the second ramp is the same as the speed of the combined mass immediately after the collision of the two bodies at the bottom of ramp 1.
Problem 5: In part (e), first obtain the radius r_0 for a circular orbit. Then let r = r_0+δ(t) and write down the differential equation satisfied by the deviation δ. Does this equation look familiar?
Problem 6: Euler equations are valid for general force-free rotations. For this problem, perturb the angular frequency by letting ω = (α, β, ω) where α and β represent a small perturbation about (0, 0, ω). Then find the equations satisfied by α and β. Do those equations look familiar?
Problem 7: Deceptively hard! You may want to consider the system to be made up of 3 components (glass, the ground, and "outer space"), each of which remains in equilibrium. Read the problem carefully and apply what it says faithfully!
Problem 8: This is the simplest problem; a straight-forward application of the Stefan-Boltzmann law of black-body radiation. Problem 9: Since S is a state function, you can get the change of entropy as the system goes quasi-statically from (V1,T1) to (V2,T2) by using a two-step transformation: (V1,T1) to (V1,T2) first, and then on to (V2,T2). The first step is at constant volume, and the second one is at constant temperature. In the second step, note that the internal energy of the ideal gas does not change since it only depends on temperature.
Problem 10: Start from writing an expression for ideal gas entropy generalizing the work you did while doing Problem 9. You need to have a sensible inclusion of the dependence on the number of particles in this generalization (also see Fall 2014-Part II-7). Assume first that A and B are of the same type (e.g., monatomic) of ideal gas and thus they have the same heat capacity per particle at constant volume, which would imply that TA = TB initially. Then come up with some thermodynamic argument to generalize to the case where TA and TB may be different initially.
Good luck! Hisao Nakanishi
Homework 2 Supplemental Discussion
Thu 10Sep2015 10:00AMAs I promised in class today, here are some discussions on Problems 1, 4(b), 8, and also Problem 9 from Homework 1:
Problem 1: Let us call the torque applied on cylinder 1 (far left) by cylinder 2 (middle) when the two come in contact (each spinning in the same direction initially) τ21 and the reverse (torque on 2 by 1) τ12, etc. The forces applied on 1 by 2 and on 2 by 1 due to contact are of the same magnitude and in opposite directions by the action-reaction principle. So, the torque vectors (relative to respective axes) are actually equal in both magnitude and direction.
Now, let us define the positive direction for the torque to be in alternating directions. So, say, τ21>0 if it is into the paper while τ12>0 if it is out of the paper. With this definition, τ21=−τ12 by action-reaction principle. Similarly, τ23=−τ32.
Why do we make such a strange convention? Let's see ... Now, let the angular momentum of cylinder 1 (and 3) relative to its own axis to be equal to L1 (and L3) with the same convention on the positive direction, and similarly for the angular momentum of cylinder 2 relative to its own axis be L2, with the corresponding convention for the positive direction.
Then,
ΔL1 = ∫ τ21 dt,
ΔL2 = ∫ (τ12 + τ32) dt,
ΔL3 = ∫ τ23 dt
Using τ21=−τ12, etc, then
ΔL1 + ΔL2 + ΔL3 = 0
This looks (almost) like conservation of angular momentum. However, it is in reality only an expression of action-reaction principle. As I explained in the earlier post, the angular momentum per se is not conserved in this system of 3 cylinders. Note that, with the alternating direction convention for the positive angular momentum, we have the initial L1i=Iω, L2i=−Iω, and L3i=Iω. Now you should be able to carry it the rest of the way.
Problem 4, part (b): The equation of motion you get while you solve for part (a) is
(d2/dt2)θ=-(3g/2L)cos θ. This can be integrated to yield
[(d/dt)θ]2 = − (3g/L)sin θ + const
where the const above equals (3g/2L) by using the initial condition. Therefore,
(d/dt)θ = − [(3g/L)(1/2-sin θ)]1/2 = 1/(dt/dθ)
Use this to write T (the time it takes for θ to change from 30 degrees to 0) as an integral over θ. The way we got the velocity as a function of displacement from the equation of motion and the way we took advantage of dx/dt= 1/(dt/dx) to write time t as a function over x are both fairly common techniques in doing something analytically for the few non-linear problems that allow them.
Problem 8: In part (a), in order to get each element Iij of the moment of inertia tensor, you subtract the contribution from the bit of mass at the old location and add it at the new location. This changes I11, I33, I23, and I32 to first order in Δa. Then L = {I} ω gives L to first order in Δa. You will see that L now has a y-component when ω only has a z-component. So they are not in the same direction!
Now part (b) to get the angle α between L and ω, you can calculate the scalar product L⋅ω = ω L cos α. There is a slight problem, however. Since cos α = 1-(1/2)α2... for small α, you needed to calculate the scalar product (and thus {I}) to order (Δa)2 in order to calculate α to order Δa. Unfortunately, in part (a), however, you were asked to calculate {I} only to order Δa. So, if you only use your answer to part (a) here, the angle α will come out to 0 even though L and ω are clearly not colinear.
Homework 1, Problem 9: The process of expansion into vacuum described here is NOT reversible, even though it is adiabatic and both its initial and final states are in equilibrium. So, e.g., we do NOT have PVγ=const. Instead, we have the energy conservation only during the process. So, (1) you can equate the decrease in the gas's internal energy to the increase of the potential energy of the spring, (1/2)kx2. Then, (2) the final pressure P1 times the cross sectional area of the piston V0/x to kx for the ideal spring. This allows you to eliminate x and get a relationship among P1, V0, T1, and T0. Now, use the ideal gas law for the final state to eliminate P1 and V0 and get the relationship between T1 and T0. After that, use the ideal gas law for the initial state to get the relationship between P1 and P0.
The answers: T1 = (6/7) T0, P1 = (3/7) P0. Some of you got T1 (P1) to be higher than T0 (P0). You should have recognized immediately that such answers could not be correct. Why?
Homework 2 Pointers (beware some spoilers)
Sat 05Sep2015 9:18AMHW2 collects problems having to do with rotation (even the thermal problem #10), but they do involve different kinds of "rotation". Unfortunately, we don't have the Monday class due to Labor Day holiday - so we have no opportunity to discuss in class concepts and issues involved in these problems. Therefore, I provide a fairly detailed points below. Some are spoilers - so I would suggest you try the problems without reading these comments first - and then read them only as needed.
Problem 1. This is not really a problem of angular momentum conservation in the normal sense. Why? If nothing is supporting the axles of the cylinders, they will not generally keep the same relative positions when the cylinders touch; so clearly, there is external torque that is acting on the axles relative to some selected statrionary axis, and thus the net angular momentum in the normal sense is not conserved.
However, you can discuss the problem as if it were one of angular momentum conservation but only if you define the positive direction of the angular momentum in alternately reversing directions for the adjacent cylinders. Why? Think about what (friction) forces (thus torque) act on them if two adjacent cylinders touch. The forces involved are of equal magnitude and in opposite directions (action-reaction). This gives rise to the above pseudo-conservation "theorem", which is actually only another way of implementing action-reaction principle.
Problem 2. This problem needs to specify the mass of the wedge. Let's say it is "m" (lower case). What is conserved in this situation?; when the ball rolls down, how does the wedge move? You would expect a coupled motion. Say x measures the wedge displacement; but the center of the ball moves in both horizontal and vertical directions. So what variable(s) would you use to describe its motion? Be careful with geometry!
Problem 4. Unlike the Problem 1 of Homework 1, this setup is standing vertically under gravity. The two rods are also completely equivalent unlike the two rods in the previous problem. Part (b) is an example of what one can still do analytically for a non-linear problem.
Problem 5. In all these yo-yo problems, the key is the mechanical constraints that relate the angular motion to the linear motion (and thus to the length of the string that extends).
Problem 6. This problem involves rotation and collision. When bouncing off of a surface, both linear and angular momenta change. Why? These changes can be described by using impulse and torque (which are related to each other). In the direction normal to the surface, the impulse on the spinning ball can be assumed the same as if the ball were not spinning because this is caused by the normal force upon impact. There is no need to know the details of these impulses.
Problem 8. This is the only problem here that involves a moment of inertia tensor. If you are shaky on this, review your mechanics text (e.g., Thornton and Marion, Section 11.3, or Taylor, Sections 10.1-10.3). Use the xyz-coordinate system attached to the wheel so that the origin is at the center of the wheel and z-axis is the axis of rotation of the wheel. As pictured in the problem, a small mass gets chipped off from the outer rim on one face of the wheel and relocates radially inward toward the axis (but still on the same face). In part (b), you have an example where vec(L) = {I} vec(omega) gives vec(L) that is not parallel to vec(omega).
Problem 9. Relate the changes in momentum and angular momentum of the cylinder to the impulse due to static friction. You don't know the detail of this impulse, but is that necessary?
Problem 10. Very similar to the Fall 2015 Part I Problem 8. Do you see the relationship between the two problems? To be complete, think about the density at the axis. Is it a constant or does it depend on omega? You may be surprised.
Good luck!!