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« PHYS411 Spring 2014

Homework 10 Discussions

Sun 27Apr2014 9:04AM

Problem 1: The reflection coefficient T is the absolute value squared of the ratio of the plane wave amplitudes of the reflected over the incident waves. All the other factors that go into the power flux ratio cancel between the incident and reflected waves because they are travelling in the same medium. As we discussed in class, the continuity conditions at x=0 consist of two relations. One is the continuity of the wave amplitude itself (ψ); the other relates the discontinuity of ψ' at x=0 to the mass M located at x=0. A phase of a complex number z refers to δ when you write z = r e where r>0 and δ is real. Note tanδ = Im(z)/Re(z).

Problem 2: Assume real k (no spatial attenuation), positive frequency, and that the wave is bounded as t → ∞ and it propagates to the right (toward x=∞).

Problem 3: The group velocity is dω/dk whereas the phase velocity is just ω/k (assuming ω and k are real).

Problem 4: This problem is asking you to perform a spatial (inverse) Fourier transform: ψ(x,t) = ∫ A(k) ei(ωt-kx)dk where the integral is from -∞ to +∞. To do the integral, Taylor expand ω to first order about k0.

Problem 5: For part (a), when the duration of the pulse at the source is Δt, calculate what the duration will be as the observer sees it. The transverse displacement of the source divided by this time is the observed apparent transverse velocity of the source. For part (b), consider the time dilation in the source frame.


Final Exam

Wed 23Apr2014 12:56PM

Final Exam will be on Wed., May 7, 1 - 3 pm in Rm.331.

It will nominally be cumulative. Specifically, there will be one question on rigid body (top), one question that is similar to one of the questions in Exam 2, one from Chap.13, and one basic question in relativity (Chap.14) that is related to Homework #10.

Good luck!


Homework 9 Discussions

Fri 18Apr2014 3:44PM

Problem 1: Extend the discussions in class to include the stiffness term. You may assume that the allowed variations are those that agree with the stationary solution at both ends of the string as well as both ends of time in the expression for J; that is, η(L,t)=η(0,t)=0 for all t, η(x,t1)=η(x,t2)=0 for all x, and, in addition, (∂η/∂x)(L,t)=(∂η/∂x)(0,t)=0 for all t. You will have to do integration by parts twice to take care of the stiffness term.

Problem 2: Let the longitudinal displacements be xj (j=1,...,n) and set up the equation of motion for xj similarly to the way we did for 2 or 3 coupled oscillators.

Problem 3: I haven't yet passed the loaded string solution to the continuum limit (will do on Monday) but for this problem, you can follow Example 13.1 closely. Integrations are a bit cumbersome, but don't cut corners.

Problem 4: Again, Example 13.2 already gives you the steady-state solution (using what you already know from Chap.3). You just add the tansient solution to this to make it complete. That too follows from what you know from Chap.3.

Problem 5: You may model the transmission line as a discrete set of inductors L' and capacitors C'; the inductors along one side of the transmission line (say the "supply" side) in series and capacitors between the two sides ("supply" and "return") in parallel, spaced by, say Δx (in the spirit of loaded strings in Chap.12. Then, write the "equation of motion" (in this case, an equation involving L', C', and Ij where Ij is the current through the j-th inductor). Finally, take the limit of Δ→0.


Exam II Location

Wed 09Apr2014 12:32PM

Exam II will be in Room B071 of Armstrong Hall. It is a room facing the cafeteria in the basement of Armstrong.


Exam II

Sun 06Apr2014 6:01PM

We will have our Exam II on Monday, April 14 at 7 - 9 pm in a location to be announced in class. 4 problems, closed-book as in Exam I and the exam will cover whatever is covered after Exam I (up to this Wednesday) plus one problem that is similar in content to the problems of Exam I.

 


Homework 8 Discussions

Sun 06Apr2014 5:51PM

Problem 1: In (b), the main thing is to see if increasing the amplitude A increases or decreases the period T. To do this without actually doing the integration, you must change the variable of integration to put all dependence of T on A in one term in the epression for T. You are to then confirm your predictions numerically in (c). So, to "solve" in (c) refers to calculating and plotting numerically x(t) as a function of t for at least several periods (as I demonstrated for the physical pendulum problem in class). You only need to show the results (not the computer program used) and you may use any programming language and visualization method you are comfortable with.

Problem 2: We will discuss iterative maps in class and the related graphical method to find the roots of equations. Basically, you solve f(x)= g(x) by drawing the graphs y1=f(x) and y2=g(x) and look for their interscrtion where y1 = y2. As we will find in class, you have to make a zig-zag iteration graphically, and for that to work, one of the functions must be easily invertible.

Problem 4: Note the potential energy U is related to the force by F = -(dU/dx).

Problem 5: For an iterative map xn+1 = f(xn), a fixed point x0 is where x0 = f(x0). Let x1=x0+ε, d1 = x1-x0=ε, ..., etc., so that dn = xn-x0. Then use the Taylor expansion to the lowest order.


Homework 7 Discussions

Thu 27Mar2014 11:10AM

Problem 1: Taking the displacements from the equilibrium positions as directed makes your life much easier especially in a problem like this. Very similar to the original, horizontal example we treated in class except that the connection to the external wall is only at the top.

Problem 2: The small angle approximation in this case also includes the result that the spring can be assumed to be essentially horizontal, as well as tan θ ≅ θ, etc. Remember that the normal coordinates are P-1 times the column vector of the two angles where P is a matrix made of the 2 normalized eigenvectors as its columns. Here, P is unitary and thus the inverse is just its Hermitian conjugate.

Problem 3: As I explained in class, the rails are stationary and only the beads move. Use trigonometry judiciously so you avoid using square roots in the equations of motion. Note, the two modes you get actually reverse their descriptions (sym or anti-sym) if you rotate the picture by 90 degrees.

Problem 4: Combine the usual procedure for the coupled oscillators with what we discussed on how to incorporate a dissipative term into the equations of motion.

Problem 5: Set up Kirchoff's equations. The signs are very important.

Problem 6: Suitable variables for the platform is its displacement (from its equilibrium position) and the angle from vertical for the pendulum. The two oscillate and are coupled because the pendulum is hang from a point fixed relative to the platform. If the equations of motion (or Euler-Lagrange equations) are set up correctly, the result should also keep the center of mass to remain stationary (or moving with constant velocity if the initial condition was such).

Problem 7: The zero frequency mode is analogous to the tri-nuclear molecule we discussed in class or Problem 6 above.


Homwork 6 Submission

Mon 24Mar2014 2:13PM

On Homework 6 (and only on this homework), I will accept late submission by up to two days (48 hours), but with a penalty. If a late HW6 is submitted to me by noon on Tuesday, 3/25, it will be subject to 10% penalty, and if it is submitted in class on Wednesday, 3/26, it will be subject to 20% penalty. This is a special arrangement in view of the fact that some unforseen events might have happened to you during the spring break. Please understand that the original deadline (printed on the Homework itself) was the Wednesday prior to the spring break. It had been postponed already by your request because of the (then) upcoming spring break. Thank you.


Homework 6 deadline postponed till Monday class on March 24 (after you come back from Spring Break).

Mon 10Mar2014 12:30PM

As it says in the title.

Homework 7 will be issued on Monday, March 24 as well, containing a couple more problems as usual.


Homework 6 Discussions

Sun 09Mar2014 9:58AM

Problem 1:Maybe deceptively difficult. For starters, you need to consider the prolate (I > I_3) and oblate (I < I_3) cases separately.

Now, draw the space and body cones (cf. discussions in class and text) so that the top (base disk) of the two cones are flash and the line where the two cones touch represents the angular velocity vector ω. For ω to be coplaner with the net angular momentum vector L (along fixed x'_3 axis) and the body x_3 axis and for it to circularly precess around x_3 axis at constant angular frequency, the body cone must roll without slipping on the space cone.

For part (a), write the top radii of the cones in terms of ω and the angles given using trigonometry. For (b), use the fact that the body cone rolls around the space cone without slipping. This means that there is a certain relationship between Ω,r and Ω',R. Then, you also have tanθ/tanα=I/I_3.

Problem 2: Draw the cones as in Problem 1, but now at t=0 the x_3 axis is vertically upward. Part (a) is a geometry problem where you need to find what the maximum angle ω can make with x_3 as it rolls around. This max angle would be 90º if ω barely touches the horizontal plane.

Problem 3: More than one way to do this. One way is to use the result of Homework 3, Problem 4. Another way is to build on our class discussion of using the Eulerian angles to represent a 3D rotation.

Problem 4: For any rotation, the axis is an eigenvector with an eigenvalue of 1. Any rotation can be represented as one about the z-axis by first transforming the axes, i.e., R'=U^{-1}RU, where R' is now a rotation about the z-axis. In this representation, the amount of rotation γ is related to the trace of R' in a simple way. But the trace is invariant under a similarity transformation!

Problem 5: Euler equations with torque. Solve for ω_3 first, and then work on the other components.

 


Grader Office Hour cancelled

Thu 06Mar2014 2:21PM

Our grader, Mr. He, reports that not a single student has taken advantage of his office hours so far. Under these conditions, it is unreasonable to require him to keep such hour, and I am suspending it as of now, until further announcement. If you would like to have it reinstated and would use it it were to happen, please let me know so that we could act on that. Thank you.

H. Nakanishi


Homework 5 Discussions

Sat 22Feb2014 2:16PM

Please do not forget this homework set though the first midterm exam is also coming up on Thursday. Doing this homework thoroughly will be useful for (at least part of) the upcoming exam.

Problem 1: This is by far the hardest problem in this set. Take the radius R to be that of the circle drawn by the contact point between the coin and the horzontal surface. Consider the fixed frame origin O' to be at the center of this circle with z'-axis pointing vertically up. Then, the body xyz-frame is to be centered at O, the center of the coin, and z-axis through the coin's axis (say, pointing outward) while x- and y-axes are in the plane of the coin. Then the x, y, and z-axes are the principal axes of the coin. First, calculate the principal moments of inertia for the coin.

The idea is to write down the torque tau = (d/dt) L (angular momentum), but how do you obtain tau and L?

The torque tau (about O) is due to the contact forces on the coin, i.e., friction and normal force. The friction provides the needed centripetal force on the coin to execute the orbital motion - don't forget that the radius of this orbital motion of the coin's center of mass is less than R (because of its tilt from vertical).

L is the total angular momentum (spin plus orbital) written in the body coordinates. Though L has more than one component, you will see that tau points only in one of the body coordinate directions. Thus, only a part of L is changing in time. The precession rate of this part is the angular frequency of the coin's orbital motion.

Problem 2: Follow the discussion of Example 11.11 in Thornton and Marion. Since this is a torque-free case, the angular momentum is constant. Thus it is best to take z'-axis of the fixed frame to be along L. Then use the fact that L (z'-axis), omega, and z-axis are in the same plane all the time.

Problem 3: We discussed the rotation of the coordinate system and how that is reflected in the moment of intertia tensor at great length in class.

Problem 4: Just go back to the definition of the moment of inertia tensor.

Problem 5: Use a suitable (and different) body coordinate system for each case. For the case where an edge of the cube is fixed on the table, clearly, the midpoint of that edge works well as the origin. For the case where there is no friction, the cube's center of mass has no horizontal motion. So in that case, the CM can serve well as the origin. Then, the first thing you should do is to find the suitable moment of inertia tensors. The case without friction will turn out to be the more difficult, since you have to treat the normal force very carefully in that case. In both cases, once you come up with the equation of motion (tau = (d/dt) L) , you can't solve it in closed form as it will be non-linear. However, it should be possible to cast it exactly in the form of (angular velocity) = function of the tilt angle theta. Or, alternatively, you can use energy conservation.

 

 


Exam I

Fri 21Feb2014 5:20PM

First Midterm Exam has been scheduled. It will be on Thursday, Feb. 27, 7 - 9 pm in RM.333. It will cover evertything treated in the course by the Wednesday class, Feb. 26 - i.e., the tides portion of Chap.5, Chap. 10, and parts of Chap. 11 up to material covered on Feb.26 and corresponding homework, class discussions, reading from the text, and discussions presented on this site.


Homework 4 Discussions and first Midterm Exam

Sun 16Feb2014 5:01PM

I would like to give the first of the two Midterm Exams as a 2-hour evening exam during the week of Feb. 24. Please think about your schedule for that week and be ready to be polled in classes this week. The exam will cover materials from the course beginning to whereever we will be right up to the class just prior to the exam.

Now about Homework 4.

Problem 1: Part (c) gives an example where the angular momentum L and the angular velocity vectors are not in the same direction. Get the angle as the inverse cosine, e.g., of the dot product of the two (divided by the product of their magnitudes).

Problem 2: One way to calculate a (scalar) moment of interia I' about an arbitrary axis e is is to rotate {I} to {I'} (or equivalently the coordinate system xyz to x'y'z') so that e is along the new z' direction. Then, the wanted I' is the I'_{33} element of the {I'} tensor. Then what you need to know is what this required rotation means. Let {I'} = U^{-1} {I} U. Since {I'} acting on the column vector (0,0,1) should correspond to {I} acting on the axis e, we see that U acting on (0,0,1) column vector = e.


Homework 3 Discussions

Fri 07Feb2014 1:37PM

Unusual weather problems have led to headaches in our PHYS 411 scheduling. I apologize about the unexpected bumps in lecturing and homework. I hope all will be well now - am holding my breath!

Please raise issues, ask questions, or request help/suggestion if you have difficulty with any aspect of this course. Do not keep to yourself if you have problems.

Problem 1: The rotating frame (x,y,z) is the one moving on a circular track with the bicycle (but NOT spinning with the wheel). Then use the usual relationship between the acceleration in the fixed frame and that in the rotating frame. Express all terms in terms of the coordinates in the fixed frame for this problem.

Problem 2: This is probably the most difficult one by far in this set. All the O(\omega^2) corrections have dominant contributions of the form \omega^2 (h^2/g) \sin lambda \cos lambda times a number. Further corrections for the same order in \omega (if you are so inclined to think about them) are smaller than these by at least a factor of h/R. The "number" in front are the C_i Thornton and Marion want you to calculate.

(a) The omega^2 contribution from the Coriolis force should be easy. The easterly Coriolis acceleration resulting from vertical fall of the mass (which is of order omega) produces an easterly component to the velocity. If you feed this back into the Coriolis acceleration, you get an order \omega^2 southerly velocity. This latter will result in a southerly deflection.

(b) The total centrifugal acceleration is -\omega x [\omega x (R+r)]. Thus, as the mass falls and r changes, it changes. If you take r=z, you can easily calculate the projection of this acceleration in the southerly direction. The "R" term leads to the baseline (plumb line) and the "z" term gives rise to the correction you seek.

(c) This is the difficult part. The actual gravity acting on the mass (i.e., toward the center of Earth) varies as the mass falls both in magnitude and in direction. It is easy to see why the magnitude varies, but the direction? The direction changes because the falling trajectory is NOT vertical (neither is it exactly in the "plumb line" direction when you define the plumb line to be the line made by the starting point of the fall and the point where the plumb bob touches the ground for the plumb line that is just long enough to do so).

You must first recalculate the fall time. Because of the changing magnitude of the gravity, this is no longer just (2h/g)^(1/2) but acquires a correction term. The height of the falling particle z(t) is also not exactly h-(gt^2/2) any longer as well. These corrections must be fed back in to the calculation of deflection that uses the centrifugal acceleration -\omega x [\omega x (R+r)]. You will see that only the "R" contribution acquires the dominant correction due to the correction in the fall time. These considerations then lead to (5/3) for a part of C_3.

Now the change in the direction of g as the mass falls. This actually has two separate effects. One is to redefine the baseline shift (or the location where the plumb bob touches the ground). The other is to add a correction term to the centrifugal acceleration that you integrate twice. The total correction from these adds up to (5/6) for the remainder of C_3. Thus C_3 = 5/2 in total. This part of C_3 is often missed in calculations you might find in literature that claim Thornton and Marion to be wrong.

Problem 3: Other than being on the southern hemisphere, this is very similar to what we already did.

Problem 4: We will be talking about 3D rotations at some length in Chapter 11. This problem is another way of deriving the transformation dealing with a general rotation in 3D, a different way from the one that uses Eulerian angles or the one that uses so-called directional cosines which we will discuss in class.

First, rotating the coordinate system requires the same transformation as inversely rotating a vector in the original coordinate system. Considering the latter instead may be simpler. Second, rotating a vector about an axis can be performed by first decomposing the vector into the components along and orthonal to the axis of rotation. The axis component then will be invariant under the rotation. The orthogonal component will rotate the same way as a 2D vector lying on the plane perpendicular to the axis.

Problem 5: We will define the 3D moment of inertia tensor on Monday. The principal axes and moments refer to the eigenvectors and eigenvalues of the 3x3 tensor you get in part (a). We will discuss the meaning of these quantities on Monday and later. These are some of the main topics of Chap.11. For this problem, you just need to know the definition of the tensor and how to diagonalize (or get eigenvalues/eigenvectors) of a matrix.

 

 

 


Homework 2 Discussions

Fri 31Jan2014 10:32AM

Please read and take what I say below as you need it. Don't come to depend on these hints; it's there just to help you get unstuck if you are in fact stuck.

Problem 1: Assume that the sea surface is tilted vertically relative to the Earth's surface so that it makes an angle with the east-west line. The leading physics here is that, in the east-west-vertical plane, there is a force balance between gravity, Coriolis force, and a normal force (perpendicular to the water surface). This gives you the tilt angle. Then you know how much of a water level difference there is over the 100 km between Miami and the Bahamas Grand Bank. If there were no Coriolis force, then there would be no such level difference.

Problem 2: As in Problem 2, the net force due to pressures from surrounding fluids and air must be normal to the fluid surface.

Problem 3: As I mentioned in class briefly, when you throw a particle upward and wait for it to come back down, the Coriolis deflection does not cancel on the way up and on the way down. Why?

Problem 4: True Lagrangian L must be calculated in an intertial (our "fixed") frame of reference. But then such an L can be re-expressed in terms of quantities as measured in a rotating (or non-inertial) frame of reference. In this problem, you are asked to treat such a quantity (say, L_r, which is equal to L but written in terms of rotating frame quantities) as if it were a true Lagrangian in those rotating frame quantities and obtain the corresponding momenta and Hamiltonian H_r. Then take it further by assuming that the action from L_r is stationary in the variations of (r, v_r) to come up with the corresponding Euler-Lagrange Equations. This procedure is justfied since the resulting Euler-Lagrange Equations are indeed the correct equation of the dynamics (i.e., 10.25).

Problem 5: We will briefly discuss the Roche limit in class on Monday. Basically, you simply apply the analogy of the planet to the Moon, the rock to the Earth, and the sand to water in the ocean tide discussion we already made. If the sand were only attached to the rock by gravity, then it would fly away if the tidal force beats the direct gravity which binds it to the rock. This idea of Roche limit is very important both on the scale of satellites and on much larger astronomical scales such as a new born star at a galactic center.

Problem 6: This is a problem of doing the Newton's well by equating the pressure at the center of the Earth calculated in two different ways: one with cumulative force (tidal and direct gravity) in the direction of maximum tidal force (say, x-direction) and the other the corresponding force in the perpendicular direction (y-direction). The text calculation was similar but a bit different since it calculated gravitational potential energy difference by integrating tidal force (only) over distance.

Problem 7: Note that the Moon rotates about the Earth in the same direction as the Earth revolves about its own axis.

Good luck!  Please don't leave anything unclear and un-understood. If you have not understood every detail of the previous homework, please take steps to remedy that. Go talk to the grader (Mr. He, office hours, Fri. 11:30 - 12:30 in Rm.31), me, and/or your colleagues. If you need help or clarification on current problems or any of the class discussions, please feel free to knock on my door too.

H. Nakanishi

 


Homework 1 Discussions

Mon 20Jan2014 10:29AM

There may be some spoilers below; so only read if you need some pointers or clarifications:

Problem 1: Consider the problem in the frame attached to the accelerating car. The frictionless pivot cannot provide torque but only a force against pushing it into the roof, i.e., rod's length-wise force. Also be careful about the signs (consider carefully what kind of equilibrium can be possible). Assume that the initial condition for (b) is phi=phi_0, and rod is at rest (relative to the car) at t=0.

Problem 2: Recall that the direction of the static friction force is what is needed for it to balance the other forces when the object remains at rest (relative to the merry-go-round frame).

Problem 3: Assume that the g_eff makes an angle of lambda (same as the northern lattitude) with the equatorial surface. If you love challenges, you might consider how much error is madeby this assumption (since g_eff does not really point toward the center of the Earth).

Problem 5: If you can solve Problem 2, you should not have a problem with this one.


Mr. Liang He's Office Hour

Fri 17Jan2014 11:09AM

Our grader Mr. Liang He will have his office hours weekly on Fridays at 11:30 am - 12:30 pm in Rm.31. This starts today! Occasionally, he may shift the office hour, but he will inform the students in that case.


Welcome

Tue 14Jan2014 3:19PM

Welcome to the PHYS411 students! This will be the sole online source of communication and information at least from my side. You may Email me or call me or come to my office to discuss anything related to the course, but everything that I need to inform the class online will be placed here.

Right now, the only thing here is the syllabus. However, I will be placing various stuff here as the time goes on.

For the present discussion of dynamics in non-inertial frames of reference, IF the discussions in class and in your assigned text are leaving you a bit unsettled, you might enjoy the corresponding discussions given in some other books, such as Taylor or Kleppner and Kolenkow (references 4 and 8 on the syllabus). They give somewhat different tacks in explaining the key concepts.


Last Updated: Jun 1, 2016 10:24 AM