Physics 351 Monday, April 3, 2017

Transcription

1 Physics 351 Monday, April 3, 2017 This weekend you read Chapter 11 (coupled oscillators, normal modes, etc.), but it will take us another day or two to finish Chapter 10 in class: Euler angles; Lagrangian description of symmetric top; space cone, body cone, etc. for torque-free symmetric top. We ll do a quiz this Wednesday (HW9), since we ve skipped 2 weeks in a row. FYI intuitive description of precession:

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3 Torque-free precession of symmetric top (more on this later): As seen from body frame, ω precesses about ê 3 with frequency Ω. As seen from the body frame, what does L do? What does the situation look like from the space frame?

4 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 a frequency that takes some effort to calculate. (You ll calculate the space-frame precession frequency in HW11/problem 3.)

5 Video from two 2015 students traveling back from spring break: Astronaut version: Cosmonaut version (!): Dzhanibekov effect Someone s quasi-intuitive explanation: the-dzhanibekov-effect-an-exercise-in-mechanics-or-fiction-explain-mathemat

6 When we speak of L, we speak of it as calculated in the inertial space frame. Of course, according to an observer in the body frame, the body is not spinning, but since we mean the inertially calculated value L, an observer in the body frame still sees this vector change direction, so it has a nonzero value and a nonzero derivative. The body frame at any given time relates to the space frame through the Euler angles. (On a separate topic:) I d love to see a walkthrough on the Euler axes just to be sure I understand. When we talk about L, we almost always mean L as calculated in a non-rotating frame. We can then at any time t project the vector L onto the instantaneous ê 1, ê 2, ê 3, axes. This is what we do if we want to use the Euler equations, which refer to the ê 1, ê 2, ê 3 projections of the ω and L that we calculate in the space frame. One notable exception was the coin in HW10/q8. In that case, we had our choice of calculating L in the lab frame or in the turntable frame. Either way we calculated L, we would find ( ) ( ) dl dl = + Ω L dt dt lab frame turntable frame

7 One useful tool for relating the fixed ˆx, ŷ, ẑ axes to the rigid body s ê 1, ê 2, ê 3, axes is the Euler angles, φ, θ, ψ. (Another way, which I used in the simulation program for the struck triangle, is simply to keep track instant-by-instant of the x, y, z components of ê 1 (t), ê 2 (t), ê 3 (t). But if you re given the three Euler angles, you can compute the x, y, z components of the body axes ê 1, ê 2, ê 3.) Question: Suppose I rotate the vector (x, y) = R(cos α, sin α) by an angle φ (about the origin). How would you write x as a linear combination of x and y? How about y as a linear combination of x and y?

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9 Mnemonic: for infinitessimal rotation angle ɛ 1, r r + ɛˆn r. So for rotation about ŷ, (1, 0, 0) (1, 0, ɛ), since ɛŷ ˆx = ɛẑ.

10 The hardest part of writing down 3 3 rotation matrices is remembering where to put the minus sign. Once you ve worked out one case correctly (e.g. from a diagram), here s a trick (thanks to 2015 student Adam Zachar) for working out the other two...

11 Just add two more columns and two more rows, following the cycles: xyz, yzx, zxy. Then draw boxes of size 3 3.

12 (Check previous result using Mathematica.)

13 Euler angles: can move (x, y, z) axes to arbitrary orientation.

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15 Suppose the Euler angles φ, θ, ψ vary with time, as the body orientation changes w.r.t. the space frame s fixed x, y, z axes. I ll write out more steps than Taylor does, and I may confuse you by saying (ˆx, ŷ, ẑ) (ê 1, ê 2, ê 3 ) (ê 1, ê 2, ê 3 ) (ê 1, ê 2, ê 3 ). I do this so that my (ê 1, ê 2, ê 3 ) are the same as Taylor s. 1. Rotate by φ about ẑ ê 1, ê 2. (ê 3 = ẑ.) 2. Rotate by θ about ê 2 ê 1, ê 3. (ê 2 = ê 2.) 3. Rotate by ψ about ê 3 ê 1, ê 2. (ê 3 = ê 3.) ω = φ ẑ + θ ê 2 + ψ ê 3 = φ ẑ + θ ê 2 + ψ ê 3 Next, project ω onto more convenient sets of unit vectors. Note: ω is a vector, but ω is not the rate of change of any vector. Infinitesimal rotations commute, but finite rotations do not.

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17 In the space basis [proof on previous page]: ω = ( θ sin φ+ ψ sin θ cos φ)ˆx + ( θ cos φ+ ψ sin θ sin φ)ŷ + ( φ+ ψ cos θ)ẑ In the body basis [proof on next page]: ω = ( φ sin θ cos ψ+ θ sin ψ)ê 1 + ( φ sin θ sin ψ+ θ cos ψ)ê 2 + ( φ cos θ+ ψ)ê 3 Most convenient for symmetric top (λ 1 = λ 2 ): in the primed basis (i.e. before the final rotation by ψ about ê 3 ). Note that ê 3 = ê 3. ω = ( φ sin θ)ê 1 + ( θ)ê 2 + ( φ cos θ + ψ)ê 3 This last one is easiest to see if you consider the instant at which ψ = 0.

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19 Most convenient for symmetric top (λ 1 = λ 2 ): ω = ( φ sin θ)ê 1 + ( θ)ê 2 + ( φ cos θ + ψ)ê 3 Now, how do we write the top s angular momentum L and kinetic energy T? How about the Lagrangian?

20 Most convenient for symmetric top (λ 1 = λ 2 ): ω = ( φ sin θ)ê 1 + ( θ)ê 2 + ( φ cos θ + ψ)ê 3 Now, how do we write the top s angular momentum L and kinetic energy T? How about the Lagrangian? L = ( λ 1 φ sin θ)ê 1 + (λ 1 θ)ê 2 + λ 3 ( φ cos θ + ψ)ê 3 T = 1 2 λ 1( φ 2 sin 2 θ + θ 2 ) λ 3( φ cos θ + ψ) 2 L = 1 2 λ 1( φ 2 sin 2 θ + θ 2 ) λ 3( φ cos θ + ψ) 2 MgR cos θ Which coordinates (Euler angles) are ignorable? What are the corresponding conserved momenta?

21 ω = ( φ sin θ)ê 1 + ( θ)ê 2 + ( φ cos θ + ψ)ê 3 L = ( λ 1 φ sin θ)ê 1 + (λ 1 θ)ê 2 + λ 3 ( φ cos θ + ψ)ê 3 L = 1 2 λ 1( φ 2 sin 2 θ + θ 2 ) λ 3( φ cos θ + ψ) 2 MgR cos θ

22 L = 1 2 λ 1( φ 2 sin 2 θ + θ 2 ) λ 3( φ cos θ + ψ) 2 MgR cos θ

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24 (Skip: Just in case you wanted to see the θ EOM derived.) L = 1 2 λ 1( φ 2 sin 2 θ + θ 2 ) λ 3( φ cos θ + ψ) 2 MgR cos θ

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27 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.)

28 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.

29 (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.

30 (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.

31 (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

32 (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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38 Physics 351 Monday, April 3, 2017 This weekend you read Chapter 11 (coupled oscillators, normal modes, etc.), but it will take us another day or two to finish Chapter 10 in class: Euler angles; Lagrangian description of symmetric top; space cone, body cone, etc. for torque-free symmetric top. We ll do a quiz this Wednesday (HW9), since we ve skipped 2 weeks in a row. FYI intuitive description of precession: