Physics 351 — Wednesday, February 25, 2015

I Quiz #4 (a HW4 problem) for last 15 min of class today. Onesheet of your own handwritten notes is OK.

I Be sure to show your work on quizzes. Several of you got 4/5on Quiz 3 for a correct answer whose steps I couldn’t follow.

I Remember HW6 due Friday in class. Handing out HW7.I Study/help session times/locations:

I Wed. 5–7pm, DRL 2N36 (Bill subs for Tanner this week)I Thu. 4–10pm, DRL 3C2 (Bill 4-6pm)I I’m normally in or near DRL 1W15, 9am-6pm. You’re welcome

to 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 can still get full credit if you turn in with HW6 any XCproblems from HW4/5 that you haven’t done yet. Just makeit obvious to Tanner what problem you’re solving.

I Anyone interested in animating HW5/q11 motion (or otherinteresting HW5 scenarios) for extra credit?

I We left off Monday discussing last weekend’s reading Q’s.

(2) Why is Ueff(r) non-monotonic, unlike U(r) = −Gm1m2/r?Is the time for r to oscillate back and forth between rmin and rmax

always equal to the time in which φ advances 360?

For inverse-square-law forces (U ∼ −1/r) and for (isotropic)Hooke’s-law forces (U ∼ r2), the period of the φ motion equalsthe period of the r motion, and the orbit always closes on itselfafter one revolution. For more general U(r), the orbit does notnecessarily repeat itself (non-closed orbit).

(inverse-square force) (more general case)

(2) Why is Ueff(r) non-monotonic, unlike U(r) = −Gm1m2/r?Is the time for r to oscillate back and forth between rmin and rmax

always equal to the time in which φ advances 360?

For inverse-square-law forces (U ∼ −1/r) and for (isotropic)Hooke’s-law forces (U ∼ r2), the period of the φ motion equalsthe period of the r motion, and the orbit always closes on itselfafter one revolution. For more general U(r), the orbit does notnecessarily repeat itself (non-closed orbit).

(inverse-square force) (more general case)

(Taylor 8.12. Would have been on HW7, but let’s do it in class.)

(a) By examining d/dr of the radial effective potential

Ueff(r) = − Gm1m2

r+

`2

2µr2

find the radius r0 at which a planet with angular momentum ` canorbit the sun in a circular orbit with fixed radius.

(b) Use d2Ueff/dr2 to show that this circular orbit is stable, i.e.that a small radial nudge will cause only small radial oscillations.

(c) Show that the frequency Ω of these radial oscillations equalsthe frequency ω = φ of the planet’s orbital motion.

We started out with

L =12m1r

21 +

12m2r

22 − U(|r1 − r2|)

Then using CM coordinate R (and r = r1 − r2) let us write

L =12MR2 +

12µr2 − U(r)

making R ignorable. Working in (inertial) CM frame let us write

Lrel =12µr2 − U(r)

Angular-momentum conservation keeps motion in fixed plane, so

L =12µ(r2 + r2φ2)− U(r)

making φ ignorable: the φ EOM is just

µr2φ ≡ `

L =12µr2 +

12µr2φ2 − U(r) µr2φ ≡ `

Radial EOM is (you can’t plug in φ = `/(µr2) before this):

µr = −dUdr

+ µrφ2 = −dUdr

+`2

µr3= − d

dr

(U(r) +

`2

2µr2

)where the “effective potential” is U + “centrifugal potential”

Ueff(r) = U(r) +`2

2µr2

Now energy conservation lets uswrite

E ≡ 12µr2 + Ueff(r)

E =12µr2 +

`2

2µr2+ U(r)

So r just oscillates back and forth atconstant E.

Radial equation of motion:

µr = −dUdr

+`2

µr3= F (r) +

`2

µr3

The trick to solving the radial EOM is to substitute u = 1/r.

u′′(φ) = −u(φ)− µ

`2 u(φ)2F

(which you’ll use in HW7 next week).

If you plug in F = 0 (no force), you get u′′(φ) = −u(φ) whosesolution is

r(φ) =1

u(φ)=

r0

cos(φ− δ)which you never would have believed was an equation for a straightline if I hadn’t made you show, in HW4/q7, that in polarcoordinates the shortest path between two points in a plane isdescribed by r cos(φ+ α) = C.

Back to the “transformed radial equation”

u′′(φ) = −u(φ)− µ

`2 u(φ)2F

If we specialize to (attractive) inverse-square forces (γ = Gm1m2):

F (r) = −Gm1m2

r2= − γ

r2= −γu2

then the u2 cancels, making the last term a constant

u′′(φ) = −u(φ)− γµ

`2

The solution to (u+ const)′′ = −(u+ const) is that(u+ const) is a cosine:

u(φ) =γµ

`2(1 + ε cosφ)

r(φ) =1

u(φ)=

`2/(γµ)1 + ε cosφ

=`2/(Gm1m2µ)

1 + ε cosφ≡ c

1 + ε cosφ

So for the attractive inverse-square case F = Gm1m2/r2 we find

r(φ) =`2/(Gm1m2µ)

1 + ε cosφ≡ c

1 + ε cosφ

where c has dimensions of length and the dimensionless parameterε is the eccentricity.

As you know, this equation can describe a circle, an ellipse, aparabola, or a hyperbola. We’ll come back to that on Friday.

r(φ) =c

1 + ε cosφ

(Taylor 8.19) The height of a satellite at perigee is 300 km abovethe earth’s surface, and it is 3000 km at apogee.

(a) Find the orbit’s eccentricity.

(b) If the orbit lies in the xy plane, with major axis in the xdirection, and with earth’s center at the origin, what is thesatellite’s height when it crosses the y axis?

Note: earth’s radius is Re = 6400 km.

Physics 351, Spring 2015, Homework Quiz #4 (2015-02-25).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.

In a crude model of a yoyo, a massless string issuspended vertically from a fixed point and theother end is wrapped several times around auniform cylinder of mass m and radius R.When the cylinder is released it movesvertically down, rotating as the string unwinds.Write down the Lagrangian, using the distancex as your generalized coordinate. Write theLagrange equation of motion and find thecylinder’s downward acceleration, x.Remember that for a uniform cylinder,I = 1

2mR2. (Show your work!)

Physics 351 — Wednesday, February 25, 2015

I Remember HW6 due Friday in class. Handing out HW7.I Study/help session times/locations:

I Wed. 5–7pm, DRL 2N36 (Bill subs for Tanner this week)I Thu. 4–10pm, DRL 3C2 (Bill 4-6pm)I I’m normally in or near DRL 1W15, 9am-6pm. You’re welcome

to 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 can still get full credit if you turn in with HW6 any XCproblems from HW4/5 that you haven’t done yet. Just makeit obvious to Tanner what problem you’re solving.

I Anyone interested in animating HW5/q11 motion (or otherinteresting HW5 scenarios) for extra credit?