Physics 351 — Wednesday, February 11, 2015

I Quiz #2 (a HW2 problem) for last 15 min of class today. Onesheet of your own handwritten notes is OK.

I Remember HW4 due Friday in class.I Study/help session times/locations:

I Wed. 5–7pm, DRL 2N36 (Tanner)I Thu. 4–10pm, DRL 3C2 (Bill until 6pm or so)I Someone else kindly agreed to swap rooms, so we have 3C2

contiguously now from 4pm-10pm on Thursdays. I’ll be there4-6pm, or until 7pm if there is interest.

I I’m normally in or near DRL 1W15, 9am-6pm. You’re welcometo stop by my office any time Tu/Th and any time after 1pmon MWF. I’m happy to work on HW problems with you.

I Today: continue working through examples of using theLagrangian approach, including a couple of things you’ll findhelpful for HW4.

I On HW4 Q8, I should have written y = B cosh((x−A)/B)instead of y = A+B cosh(x/B).

We worked this out on Monday. William requested an animation! Imade a simple animation; Noah made a better one.

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.

positron.hep.upenn.edu/p351/files/0211_springpendulum.nb

My (pretty simple) animation:positron.hep.upenn.edu/p351/files/0211_springpendulum.nb

Noah’s (much nicer) animation:positron.hep.upenn.edu/p351/files/0211_noah_springpendulum.nb

animated gif version:positron.hep.upenn.edu/p351/files/0211_noah_springpendulum.gif

You’ll work out the motion of this “spring pendulum” in the “smalloscillations” approximation on HW5 Q7. When you solve it, it willlook familiar from Monday’s class (but not identical).

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 NDSolve 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 NDSolve 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 thatrmin/r0 = 0.208 form/M = 1/10, andgraphing r(t) and 1

2πθ(t)for this case.

Here was the HW2 problem containing a similar idea:

a similar idea appears again in HW5 Q10

The trickiest part of HW4 Q11 is writing down the K.E. I usedbrute force, and got vast numbers of cancellations/simplifications.

I believe Q12 is same scenario but now the figure is a side viewrather than a top view, so there is a G.P.E. term added to U .

Try now (with your neighbor) writing down the K.E. for Q11above. The most reliable technique is to write down the positionvs. time and then differentiate. But you may be cleverer than I am.

(I’ll hide for now the slide that goes on from here to write theLagrangian and the EOM.)

HW4 XC2 is the “threesticks” generalization of thisproblem. Let’s try the “twosticks” version.

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, Spring 2015, Homework Quiz #2 (2015-02-11).15 minutes. Work alone. Closed book, one page of handwritten notes OK.

Relax! All quizzes together make up only 10% of course grade.90% semester quiz total gets full credit.

Consider a small frictionless puck perched at the top of a fixedsphere of radius R. If the puck is given a tiny nudge so that itbegins to slide down, through what vertical height will it descendbefore it leaves the surface of the sphere?

[Hint: At what value of the normal force between sphere and puckdoes the puck leave the sphere?]

Physics 351 — Wednesday, February 11, 2015

I Study/help session times/locations:I Wed. 5–7pm, DRL 2N36 (Tanner)I Someone else kindly agreed to swap rooms, so we have 3C2

contiguously now from 4pm-10pm on Thursdays. I’ll be there4-6pm, or until 7pm if there is interest.

I I’m normally in or near DRL 1W15, 9am-6pm. You’re welcometo stop by my office any time Tu/Th and any time after 1pmon MWF. I’m happy to work on HW problems with you.

I Remember HW4 due Friday in class.

I On HW4 Q8, I should have written y = B cosh((x−A)/B)instead of y = A+B cosh(x/B).