Physics 351 Friday, April 7, 2017

Transcription

1 Physics 351 Friday, April 7, 2017 Turn in HW11 today. Handing out HW12 due next Friday. The only homework you ll do on last weekend s Chapter 11 (coupled oscillators, normal modes, etc.) is solving the Lagrangian double-pendulum problem at the start of HW12 which Taylor largely works out as an example. But I think we can work through (and demonstrate?) an example or two in class of using normal modes to decompose the motion of a set of coupled oscillators coupled linear ordinary differential equations. For the rest of the semester, we ll reinforce material we ve learned thus far; we ll learn Hamiltonian mechanics; and we ll do some supplementary readings (unconnected to HW problems or exams) to introduce some related ideas. This weekend, read Taylor s Chapter 13 (Hamiltonian mechanics) and answer Q s. For Wednesday, read Morin s chapter on Hamiltonian mechanics (PDF) and answer Q s.

2 From the final exam for the course I took, fall (This turns out to be the same problem as appears in Feynman s story of the cafeteria plate that wobbles as it flies through the air.)

3 As seen from body frame, L and ω precess about (fixed) ê 3 with frequency Ω b Ω = ω 3 (λ λ 3 )/λ, where λ = λ 1 = λ 2. As seen from the space frame, ê 3 and ω precess about (fixed) L, at frequency Ω s = L/λ 1, which you ll prove in HW11/problem 3.

4 ω = ω 1 ê 1 + ω 2 ê 2 + ω 3 ê 3 symmetric top: λ 1 = λ 2 L = λ 1 ω 1 ê 1 + λ 1 ω 2 ê 2 + λ 3 ω 3 ê 3 L λ 1 = ω 1 ê 1 +ω 2 ê 2 + λ 3 λ 1 ω 3 ê 3 = ω 1 ê 1 +ω 2 ê 2 +ω 3 ê 3 + λ 3 λ 1 ω 3 ê 3 ω 3 ê 3 L = ω + λ 3 λ 1 ω 3 ê 3 λ 1 λ 1 ω = L ( ) λ1 λ 3 + ω 3 ê 3 = L + Ω b ê 3 λ 1 λ 1 λ 1 Last line proves that ω, L, and ê 3 are coplanar (for λ 1 = λ 2 ). Torque-free (10.94): ω = ω 0 cos(ω b t)ê 1 ω 0 sin(ω b t)ê 2 + ω 3 ê 3 Key trick for understanding space and body cones: decompose ω into one part that points along L and one part that points along (or opposite) ê 3. [Sign of Ω b depends on λ 1 vs. λ 3 magnitudes.]

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6 Torque-free precession of axially symmetric (λ 1 = λ 2 ) rigid body ω = L λ 1 + Ω b ê 3 with Ω b = λ 1 λ 3 λ 1 ω 3 ω = ω 0 cos(ω b t)ê 1 ω 0 sin(ω b t)ê 2 + ω 3 ê 3 Ω space = L/λ 1 points along L. Describes precession of ω (and ê 3 ) about L as seen in space frame. ( ) ( ) dê 3 L L = ω ê 3 = + Ω b ê 3 ê 3 = ê 3 = Ω space ê 3 dt λ 1 Ω body = Ω b ê 3 points along ê 3 if λ 3 > λ 1 (oblate, frisbee) and points opposite ê 3 if λ 3 < λ 1 (prolate, US football). Describes precession of ω (and L) about ê 3 as seen in body frame. ( dl dt ) body ( ) L = ω L = + Ω b ê 3 L = ( Ω b ê 3 ) L = Ω body L λ 1 λ 1 Ω space = ω + Ω body

7 (a) Show that I = I 0 and find the constant I 0. (b) Calculate L at t = (c) Sketch ê 3, ω, and L at t = 0. (d) Draw/label body cone and space cone on your sketch.

8 (e) Calculate precession frequencies Ω body and Ω space. Indicate directions of precession vectors Ω body and Ω space on drawing. (f) You argue in HW11 that Ω space = Ω body + ω. Verify (by writing out components) that this relationship holds for the Ω space and Ω body that you calculate for t = 0.

9 (g) Find the maximum angle between ẑ and ê 3 during subsequent motion of the plate. Show that in the limit α 1, this maximum angle equals α. (h) When is this maximum deviation first reached? video: watch?v=oh-dlrifo10

10 Just FYI, I put the final exam from last time (2015) online at Let s work through Problem 1 together, which is the prolate (football-like) analogue of the oblate (frisbee-like) problem you ll work out in HW12.

11 Problem 1. A uniform rectangular solid of mass m and dimensions a a a 3 (volume 3 a 3 ) is allowed to undergo torque-free rotation. At time t = 0, the long axis (length a 3) of the solid is aligned with ẑ, but the angular velocity vector ω deviates from ẑ by a small angle α. The figure depicts the situation at time t = 0, at which time ê 1 = ˆx, ê 2 = ŷ, ê 3 = ẑ, and ω = ω(cos αẑ + sin αˆx). (a) Show (or argue) that the inertia tensor has the form I = I and find the constant I 0.

12 (b) Calculate the angular momentum vector L at t = 0. Write L(t = 0) both in terms of ê 1,ê 2,ê 3 and in terms of ˆx,ŷ,ẑ. Which of these two expressions will continue to be valid into the future? (c) Draw a sketch showing the vectors ê 3, ω, and L at t = 0. Be sure that the relative orientation of L and ω makes sense. This relative orientation is different for egg-shaped ( prolate ) objects (λ 3 < λ 1 ) than it is for frisbee-like ( oblate ) objects (λ 3 > λ 1 ). (d) Draw and label the body cone and the space cone on your sketch. (e) Calculate the precession frequencies Ω body and Ω space. Indicate the directions of the precession vectors Ω body and Ω space on your drawing. Be careful with the sign of the Ω body vector, i.e. be careful not to draw Ω body when you mean to draw Ω body.

13 (f) You argued in HW11 that Ω space = Ω body + ω. Verify (by writing out components) that this relationship holds for the Ω space and Ω body that you calculate for t = 0. (g) In the α 1 limit (so tan α α, tan(2α) 2α, etc.), find the maximum angle between ẑ and ê 3 during subsequent motion of the solid. (This should be some constant factor times α.) A simple argument is sufficient here, no calculation. (h) At what time t is this maximum deviation first reached? (This problem shows that for an American-football-like object, the frequency of the wobbling motion is smaller than the frequency of the spinning motion which is opposite the conclusion that you reached for the flying dinner plate, whose wobbling was twice as fast as its spinning.)

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20 (Taylor 10.35) A rigid body consists of: m at (a, 0, 0) = a(1, 0, 0) 2m at (0, a, a) = a(0, 1, 1) 3m at (0, a, a) = a(0, 1, 1) Find inertia tensor I, its principal moments, and the principal axes.

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33 Physics 351 Friday, April 7, 2017 Turn in HW11 today. Handing out HW12 due next Friday. The only homework you ll do on last weekend s Chapter 11 (coupled oscillators, normal modes, etc.) is solving the Lagrangian double-pendulum problem at the start of HW12 which Taylor largely works out as an example. But I think we can work through (and demonstrate?) an example or two in class of using normal modes to decompose the motion of a set of coupled oscillators coupled linear ordinary differential equations. For the rest of the semester, we ll reinforce material we ve learned thus far; we ll learn Hamiltonian mechanics; and we ll do some supplementary readings (unconnected to HW problems or exams) to introduce some related ideas. This weekend, read Taylor s Chapter 13 (Hamiltonian mechanics) and answer Q s. For Wednesday, read Morin s chapter on Hamiltonian mechanics (PDF) and answer Q s.