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Homework 10

# Homework 10 - Physics 601 Homework 10   Due Friday...

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Unformatted text preview: Physics 601 Homework 10   Due Friday November 19 Goldstein 5.8, 5.14 1. Consider a rigid body with the following mass density: ȹ x 2 + y 2 + z 2 + xy ȹ ρ( r ) = ρ 0 expȹ − ȹ 2l 2 ȹ Ⱥ a. Find the moment of inertia tensor. b. Find the three principal moments of inertia. € 2. Consider a rigid body with no external torques. Use Euler’s equations to show that the energy associated with rotational motion given in the body ­fixed frame by J2 J2 J2 E = 1 + 2 + 3 is conserved. 2 I1 2 I2 2 I3 3. In discussing rotations one can associate the angle velocity vector ω a tensor given by Ω = −ω ⋅ M . € a. Starting from the properties of the generator matrices show that ω i = 1 ε ijk Ω jk . 2 € € We know that ω is a vector (i.e. that is that it transforms like a vector under rotations). The purpose of the remainder of this problem is to use this to € demonstrate that Ω is constructed to be a tensor. b. As a first step you will need to demonstrate that the Levi ­Civita symbol εijk € transforms like a rank ­three tensor under rotations: ε'ijk = Ril R jm Rknεlmn with ε' having the same form as ε ( where the result for any rotation). (This € relatively easy once you realize that any rotation may be represented as € three rotations about fixed axes ! sing the Euler angle construction.) u € c. Using the result in b. show that " transforms like a tensor. € € 4. Consider a rigid symmetrical object with two of the principal moments of inertia equal I1 = I2 (and the third, I3 unequal) and no external torques. In this case one ! fully solve the Euler equations. Suppose that at t=0 the initial angular velocities are ( ω1 0 ), ω (20 ), ω (30 ) find ω1, ω 2 , ω 3 for all times. Discuss what your solution tells you about the frequency of the wobble in a badly thrown football (or Frisbee). € € 5. One problem with solving the Euler equations is that it gives you components € € angular velocity in the body ­fixed frame. You may want the rotation matrix (or the Euler angles) as a function of time. This problem discuss how to convert from one to another: a. Show that given ω ( body ) as a function of time R is the solution to the following t differential equation . R( t ) = R(0) − ∫ 0 dt 'ω ( body ) ( t ' )⋅ M R( t ' ) b. In general this is not trivial to evaluate. Show that the solution to this € € equation can be written in the following series form: R( t ) = R0 + R1 ( t ) + R2 ( t ) + R3 ( t )... with R0 = R(0) and t€ Rn +1 ( t ) = − ∫ 0 dt ' ω ( body ) ( t ' )⋅ MRn ( t ' ) . € € € € c. The result in b. can be written in a compact form if one introduces the € notion of time ­ordered product of two matrice swhich are functions of time: Ⱥ a( t )b ( t ' ) for t > t ' T[ a( t )b ( t ' )] ≡ Ⱥ and more generally if one has a product of Ⱥ b ( t ' ) a( t ) for t ' > t n different matrices, the time–ordered product is simply the product reorder in order descending time show that ∞ ȹ n Ⱥ 1 Ⱥȹ t ȹ − ∫ dt ' ω ( body ) ( t ' )⋅ M ȹ ȺR(0) . In showing this is sufficient for R( t ) = ∑ T Ⱥ ȹ 0 Ⱥ Ⱥ n = 0 n! Ⱥ the purpose of the problem to demonstrate that it holds for the first few terms (up to n=3). Note that the time ordering is non ­trivial since the t’ in the integral is a dummy variable and product of two integrals involves integration of two distinct dummy variable. From the form above it is common to rewrite this as a “time ­order exponential” ȹȺ Ⱥ ȹ t ȹ − ∫ dt ' ω ( body ) ( t ' )⋅ M ȹȺ R(0) . R( t ) = T Ⱥexp ȺȺ Ⱥ ȹ 0 ( ( € ) ) ...
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