Physics 351 — Monday, February 16, 2015I Some people pointed out that they would have liked to have

more time to work on the large number of XC problemsavailable in HW4, but that HW4 was already a pretty longassignment. So you can still get full credit if you turn in withHW5 any XC problems from HW4 that you haven’t done yet.Just make it obvious to Tanner what problem you’re solving.

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 You read the (*) sections of Ch7 for today; some of you alsoread Feynman’s “The Principle of Least Action” for XC.

I Today we’ll work through an example or two of using Lagrangemultipliers to solve for the forces of constraint. We’ll continuelater in the week with other ideas from the end of Ch7.

I Good Q from Friday (which I don’t yet have an answer for —maybe you do): what is the origin of the fact that the Coriolisterm does not affect the physics of HW4 problems 11 & 12?

Math 114 problem: find the point (x, y) that minimizes

U(x, y) = mg√x2 + y2

subject to the constraint y − x = 1.

Let f(x, y) = y − x− 1. Thenminimize the modified function

V (x, y) = U(x, y) + λf(x, y)

w.r.t. variables x, y, and λ.

The added variable λ is called a Lagrange multiplier.

Let f(x, y) = y − x− 1. Thenminimize the modified function

V (x, y) = U(x, y) + λf(x, y)

w.r.t. variables x, y, and λ.

Atwood machine, where (for whatever reason) you actually want toknow the tension T .

(Taylor 7.51) Write down L for apendulum in rectangularcoordinates x and y, subject to

0 = f(x, y) =√x2 + y2 − `

Write down the modified Lagrange equations.

Comparing with ~F = m~a, show that λ is (minus) the tension inthe rod.

Show that λ∂f/∂x is the component of FT in the x direction andthat λ∂f/∂y is the component of FT in the y direction.

What if instead we had written f(x, y) = x2 + y2− `2 = 0 ? Try it!

You should find that λ itself no longer equals (in magnitude) thetension, but that it is still true that λ∂f/∂x = FT,x and thatλ∂f/∂y = FT,y.

What if instead we had written f(x, y) = x2 + y2− `2 = 0 ? Try it!

You should find that λ itself no longer equals (in magnitude) thetension, but that it is still true that λ∂f/∂x = FT,x and thatλ∂f/∂y = FT,y.

(We stopped just before this page today.)(Taylor 7.52) Lagrange multipliers also work with non-Cartesiancoordinates. A mass m hangs from a string, the other end ofwhich is wound several times around a wheel (radius R, moment ofinertia I) mounted on a frictionless horizontal axle. Let x bedistance fallen by m, and let φ be angle wheel has turned.

Write modified Lagrange equations. Solve for ẍ, for φ̈, and for λ.

Use Newton’s 2nd law to check ẍ and φ̈.

Show that λ∂f/∂x = FT,x.

What is your interpretation of the quantity λ∂f/∂φ ?

HW4 XC2 was 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 — Monday, February 16, 2015

I Some people pointed out that they would have liked to havemore time to work on the large number of XC problemsavailable in HW4, but that HW4 was already a pretty longassignment. So you can still get full credit if you turn in withHW5 any XC problems from HW4 that you haven’t done yet.Just make it obvious to Tanner what problem you’re solving.

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 Good Q from Friday (which I don’t yet have an answer for —maybe you do): what is the origin of the fact that the Coriolisterm does not affect the physics of HW4 problems 11 & 12?

I Remember Quiz #3 on Wednesday: one problem from HW3.One sheet of your own handwritten notes is OK.