ph409_assign07 - Physics 409: Classical Mechanics 5 October...

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Physics 409: Classical Mechanics Fall ‘10 5 October 2010 Lecture schedule : 5 October: finish Ch. 3 (LRL vector, 2-body scattering) 7 October: Ch. 4 (rigid body kinematics, orthogonal transformations) 12 October: Ch. 4 Sec. 4.6, 4.7, 4.8, 4.9 (Skip Sec. 4.5) Reading tips for 7 October : How many degrees of freedom does a rigid body have? What are appropriate generalized coordinates? What is the difference between a space fixed coordinate frame and a body fixed frame? What is an orthogonal matrix? What are orthogonal transformations? What are they used for? What are Euler angles? Reading tips for October 12 : What is Euler’s Theorem for the motion of a body with one point fixed? What does the rate of change of a vector formula allow you to do? What is the instantaneous angular velocity vector? Homework Set #7: Due Tuesday, 12 October (1)(a) Two particles move in a central force field given by the potential V ( r ): where k and a are positive constants. Find an expression for the () 64 // V(r) k a r a r ⎡⎤ =− ⎣⎦ radius r c of a circular orbit. Show that circular orbits are possible only when the condition c # 2/3 is satisfied, where c is the dimensionless constant, . Here, l is the constant 22 /( ) cl k a μ = angular momentum, and μ is the reduced mass of the two-body system. You should obtain an analytical result for the dimensionless radius x c = r c / a that depends only on the value of c and pure numbers. (One form of the result is a quadratic equation for the variable
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ph409_assign07 - Physics 409: Classical Mechanics 5 October...

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