final-09-solutions-7

final-09-solutions-7 - Phys 12a 2009 Solutions for Final...

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Phys. 12a 2009 - Solutions for Final Exam H. J. Kimble 1. ( 15 points ) Simple oscillators - Write an equation of motion and give the frequency for free oscillation of each of the systems shown in Figure 1 below. a. 5 points b. 5 points c. 5 points B uniform magnetic field B bar magnet, magnetic dipole moment μ g g fluid of density ρ FIG. 1: Simple oscillators for Problem 1. (a) For a small range of motion, we may approximate the two-dimensional curve by a circle of radius R . The small-angle, one-dimensional motion of a ball in a circular bowl, or a bead on a circular wire, is identical to small-angle motion of a pendulum. Equation of motion: mR d 2 θ dt 2 ≈ - mgθ (1) R d 2 θ dt 2 ≈ - (2) Hence the angular frequency ω 0 is given by ω 0 = r g R . (3) (b) The for a small vertical displacement z from equilibrium, the restoring force is given by - gzAρ , where ρ is the density of the fluid and A the cross sectional
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2 area of the disk. Since the mass of the object is m 0 = hAρ 0 , where h is the total height of the cylinder and ρ 0 its density. The equation of motion is then hAρ 0 d 2 z dt 2 = - gzAρ (4) Hence the angular frequency ω 0 is given by ω 0 = r g h ρ ρ 0 , (5) with necessarily ρ > ρ 0 for the object to be floating as shown in the figure. (c) Remembering one’s elementary E+M: I d 2 dt 2 θ = τ (6) = - μ × B = - μB sin( θ ) ≈ - μBθ in the small-angle limit. Thus, ω 0 = q μB I , where I is the moment of inertia of the bar magnet.
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3 2. ( 15 points ) Energy conservation - As illustrated in Figure 2 below, a pulse ψ in ( z, t ) is propagating to the right in medium 1 and is incident upon a boundary to a second medium 2. Z 1 Z 2 ψ in ( z , t i ) ψ r ( z , t f ) ψ t ( z , t f ) z medium 1 medium 2 FIG. 2: Reflection and transmission of an incident pulse at a boundary between two media char- acterized by impedances Z 1 , Z 2 . (a) ( 5 points ) Give explicit expressions for the reflected ψ r ( z, t ) and transmitted ψ t ( z, t ) pulses from the boundary in terms of ψ in ( z, t ) and the impedances Z 1 , Z 2 for the two media. Reflection: R 12 = ψ r ( z,t ) ψ in ( z,t ) = Z 1 - Z 2 Z 2 + Z 1 Transmission: T = ψ r ( z,t ) ψ in ( z,t ) = 2 Z 1 Z 2 + Z 1 (b) ( 10 points ) Show that energy is conserved by showing that the energy transported by the reflected and transmitted pulses is equal to that of the incident pulse. From Eq. 107 in Chapter 4 of Crawford, we have that the power transported by a wave is given by P ( z, t ) = Z ∂ψ ( z, t ) ∂t 2 . (7) We must show that P in ( z, t ) = P r ( z, t ) + P t ( z, t ) , (8) or that Z 1 | ∂ψ in ( z, t ) ∂t | 2 = Z 1 | ∂ψ r ( z, t ) ∂t | 2 + Z 2 | ∂ψ t ( z, t ) ∂t | 2 . (9) Since the reflection (transmission) coefficients for the amplitude and velocity of the wave are both given by R 12 ( T ), we require that
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4 Z 1 = Z 1 | R 12 | 2 + Z 2 | T | 2 . (10) Substitute from the expressions above for R 12 and T to find that the right hand side of the previous equation reduces to Z 1 , thus confirming that the energy transported by the incident wave is equal to the sum of that for the reflection and transmitted waves.
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5 3. (20 points) Flexural waves - Consider the thin rod illustrated in Fig. 3 below. In the absence of tension (e.g., Eq. (14) in Chapter 2 with T 0 = 0), such a thin rod will
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