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a7 - (e Is the equilibrium stable or unstable 2 Consider a...

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Phys 263/Amath 261 Assignment 7 Due: Friday, July 18, 2008 (in class) 1. Consider a thin uniform circular ring of radius a and mass M . A mass m is placed in the plane of the ring. a X Y Z r r’ m φ (a) Show that the integral expression for the gravitational potential is Φ( r ) = - ρG 2 π 0 1 + ( r a ) 2 - 2 r a cos φ (1) where ρ is the appropriate mass density. (b) It makes sense, from symmetry, that an equilibrium point might exist at r = 0. Expand your result from part (a) assuming that r << a . (c) Now perform the integration over the first and second-order term in your expan- sion from part (b). (d) Find the potential energy U ( r ) and confirm that
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Unformatted text preview: (e) Is the equilibrium stable or unstable? 2. Consider a thin uniform disk of mass M and radius a . Find the force on a mass m located along the axis of the disk directly by integrating the force equation d F =-Gm dM r 2 ˆ r (2) Hint: . Use the symmetry of the problem so that you only need to consider the “vertical” (on-axis) component of d F acting on m . 3. Using Gauss’ law, calculate the gravitational field g at a distance r from an infinitely long cylindrical rod of mass density σ per unit length. 1...
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