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a6 - (c Discuss the path that the particle takes in its...

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Phys 263/Amath 261 Assignment 6 Due: Friday, July 11, 2008 (in class) 1. A central force-field allows a particle to move in a logarithmic spiral: r = ke αθ (1) where k and α are constants. (a) Find the force law associated with this orbit. (b) Determine r ( t ) and θ ( t ) for the orbit. (c) What is the total energy of the orbit? 2. A particle with mass m = 1 in straight-line motion with speed V approaches an exotic planet from a large distance. Far away, it appears as though the body will miss the planet by a distance p . The planet however exerts a central force field with strength F = - r 3 , (2) where the constant γ = 8 p 2 V 2 9 . (3) (a) Find the orbit of the particle, r ( θ ). (b) Find the distance of closest approach of the particle, and its speed at that instant.
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Unformatted text preview: (c) Discuss the path that the particle takes in its orbit, including a rough sketch. 3. Integrate directly the equation θ ( r ) = Z ( L/μr 2 ) dr r 2 μ ± E-V ( r )-L 2 2 μr 2 ² (4) to obtain the equation of motion r ( θ ) for the inverse-squared attractive force field. 4. A particle moves in an elliptical orbit in an inverse-square law central force field. If the ratio of the maximum angular velocity to the minimum angular velocity of the particle in its orbit is n , then show that the eccentricity of the orbit is ± = √ n-1 √ n + 1 . (5) 1...
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