Physics 325 Spring 2011 Problem Session 5 Solutions

# Physics 325 Spring 2011 Problem Session 5 Solutions -...

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Physics 325 Spring 2011 Discussion 5 February 21, 2011 A solid ball has a uniform mass density with total mass M and radius R . Mimic the derivation in lecture to calculate the gravitational potential Φ at a distance r > R from the center of the ball. Note that in the lecture, the potential was derived for a thin spherical shell (2-dimensional object), whereas here we have a solid ball (3-dimensional object). Start by breaking up the ball into spherical shells of radius R 0 and thickness dR 0 . Next, break up the shell into rings of width R 0 , just like in lecture. Calculate the potential d Φ due to this small ring, using the fact that - Gdm/r 0 is the potential due to a point mass dm at a distance r 0 from the mass. Then, integrate the expression over the whole ball to ﬁnd the total potential. Hint: Do the θ integral ﬁrst, by substituting u for the expression under the square root. Solution : The mass density is ρ = M 4 3 πR 3 = 3 M 4 πR 3 . We break up the ball into spherical shells, each of radius

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Physics 325 Spring 2011 Problem Session 5 Solutions -...

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