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

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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 0 ) = - ρG Z 2 π 0 q 1 + ( r 0 a ) 2 - 2 r 0 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 = 0. Expand your result from part (a) assuming that r 0 << 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 0 ) and confirm that r 0 = 0 is an equilibrium point.
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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 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 eld g at a distance r from an innitely long cylindrical rod of mass density per unit length. 1...
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This note was uploaded on 09/20/2010 for the course AMATH 261 taught by Professor Rogermelko during the Spring '10 term at Waterloo.

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