Question

Consider a very (infinitesimally!) thin but massive rod, length L (total mass M), centered around the origin, sitting along the x-axis. (So the left end is at (-L/2, 0,0) and the right end is at (+L/2,0,0) Assume the mass density λ (which has units of kg/m)is not uniform, but instead varies linearly with distance from the origin, λ(x) = c|x|.

a) What is that constant “c” in terms of M and L? What is the direction of the gravitational field generated by this mass distribution at a point in space a distance z above the center of the rod, i.e. at (0,0,z) Explain your reasoning for the direction carefully.

b) Compute the gravitational field, ~g, at the point (0,0,z) by directly integrating Newton’s law of gravity, summing over all infinitesimal “chunks” of mass along the rod.

c) Compute the gravitational potential at the point (0,0,z) by directly integrating −Gdm/r, summing over all infinitesimal “chunks” dm along the rod. Then, take the z-component of the gradient of this potential to check that you agree with your result from the previous part.

d) In the limit of large z what do you expect for the functional form for gravitational potential? (Hint: Don’t just say it goes to zero! It’s a rod of mass M, when you’re far away what does it look like? How does it go to zero?) What does “large z” mean here? Use the binomial (or Taylor) expansion to verify that your formula does indeed give exactly what you expect. (Hint: you cannot Taylor expand in something BIG, you have to Taylor expand in something small.)
e) Can you use Gauss’ law to figure out the gravitational potential at the point (0,0,z)? (If so, do it and check your previous answers. If not, why not?)

Answer 1

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