P314Final2007a-1 - P314 Classical Mechanics Sp 2007 Final...

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P314 Classical Mechanics Sp 2007 Final Complete the exam in 3 hours/ EACH PROBLEM IS WORTH 25 POINTS 1. Consider a uniform rod mounted on a horizontal frictionless axis through its center. The axle is supported by a turntable rotation about its vertical axis with angular speed Ω (see diagram). Write down Euler’s equations for rigid body motion. Choose the 1,2,3 principal axes as shown in the diagram. Find θ as a function of time by using Euler’s equations. When the rod is near horizontal show that there is simple harmonic motion in θ with frequency " = 1 2 # I 3 $ I 2 I 1 % 2. Collisions Write down the definition of differential cross section σ in terms of the angle θ through which the particles are scattered and the impact parameter b. Starting from the correct Lagrangian, prove that the scattering angle θ in any central force whose potential energy is U(r) is given by ( ) dr T r U r b r b r ! " # # # $ = % min
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P314Final2007a-1 - P314 Classical Mechanics Sp 2007 Final...

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