Physics 351 — Friday, February 10, 2017I Turn in HW4 today. (“Long and difficult but worthwhile.”) If

you’re unsure on Q11,12,13, hand in your paper after class.I I’m handing out HW5 printout (also online as a PDF). It will

be similar in length to HW4 (sorry!) but with a much higherratio of physics vs. math, so I think it will be more fun.

I Read rest of Ch7 (the (*) sections) for Monday. You can alsoread Feynman’s “The Principle of Least Action” lecture forextra credit if you like. I highly recommend it!!

I You will probably find the use of Lagrange multipliers to solvefor forces of constraint to be the trickiest part of the rest ofCh7. We will go through several examples of that next week,as it can actually be quite handy.

Did you do something like this for Q12?

One problem from HW2 (Q8) illustrated an interesting idea thatreappears this week on HW4 (Q13): work done against (or by) thecentripetal force of an object in circular motion of changing radius.

Crucial hint: the two coupled EOM can’t be solved analytically.Use NDSolveValue then FindMinimum in Mathematica.

I defined µ = m/M , let r0 = 1, then let eq1 and eq2 be the EOMfor r̈ and θ̈ respectively, in terms of µ. Then NDSolveValue tonumerically solve for r(t) and θ(t), then FindMinimum (with astarting point of t ≈ 0.01) to find r (which is same r/r0, sincer0 = 1) at its turn-around point. It’s also fun to graph r(t).

Here’s my graph of r/r0 (at turnaround point) vs. m/M (withaxis scales censored).

Checking that rmin/r0 = 0.208 for m/M = 1/10

Graphing r(t) and 12πθ(t) for the m/M = 1/10 case

Non-linear behavior is evident at large amplitude!!

a similar idea appears again in HW5 Q10

Consider a pendulum made of a spring with a mass m on the end.The spring is arranged to lie in a straight line (e.g. by wrappingthe spring around a massless rod). The equilibrium length of thespring is `. Let the spring have length `+ x(t), and let its anglew.r.t. vertical be θ(t). Assuming the motion takes place in avertical plane, write Lagrangian and find EOM for x and θ.

(We stopped here — continue Monday.)

This problem will reappear in the text of Taylor’s Ch9 (“mechanicsin non-inertial frames”), so let’s work through it by writing theLagrangian w.r.t. an inertial frame.

(7.30) A pendulum is suspended inside a railroad car that is forcedto accelerate at constant acceleration a.

(a) Write down L and find EOM for φ.

(b) Let tanβ ≡ a/g, so g =√g2 + a2 cosβ, a =

√g2 + a2 sinβ.

Simplify using sin(φ+ β) = cosβ sinφ+ sinβ cosφ.

(c) Find equilibrium angle φ0. Use EOM to show φ = φ0 is stable.Find frequency of small oscillations about φ0.

HW4 XC2 was the “threesticks” generalization of thisproblem. Let’s try the “twosticks” version. (You’rewelcome to try it for nextweek if you didn’t have timethis week. Just make itobvious to James whatproblem you’re solving.)

Two massless sticks of length 2r, each with a mass m fixed at itsmiddle, are hinged at an end. One stands on top of the other. Thebottom end of the lower stick is hinged on the ground. They areheld such that the lower stick is vertical, and the upper one istilted at a small angle ε w.r.t. vertical. They are then released. Atthe instant after release, what are the angular accelerations of thetwo sticks? Work in the approximation where ε� 1.

Now plug in, at t = 0, given conditions θ1 = 0, θ2 = ε, and findinitial angular accelerations θ̈1 and θ̈2.

Physics 351 — Friday, February 10, 2017

I Turn in HW4 today.

I I’m handing out HW5 printout (also online as a PDF). It willbe similar in length to HW4 (sorry!) but with a much higherratio of physics vs. math, so I think it will be more fun.

I Read rest of Ch7 (the (*) sections) for Monday. You can alsoread Feynman’s “The Principle of Least Action” lecture forextra credit if you like. I highly recommend it!!

I You will probably find the use of Lagrange multipliers to solvefor forces of constraint to be the trickiest part of the rest ofCh7. We will go through several examples of that next week,as it can actually be quite handy.