# Gauss' Law for Gravitation

by arl146
Tags: gauss, gravitation
 P: 343 1. The problem statement, all variables and given/known data The gravitational field g due to a point mass M may be obtained by analogy with the electric field by writing an expression for the gravitational force on a test mass, and dividing by the magnitude of the test mass, m. Show that Gauss' law for the gravitational field reads: $\Phi$ = $\oint g\bullet dA$ = -4*pi*G*M Use this result to calculate the gravitational acceleration g at a distance of R/2 from the center of a planet of radius R = 8.05 × 10^6 m and M = 8.45 × 10^24 kg. 2. Relevant equations above equation 3. The attempt at a solution i cant get the answer right for this .. heres what i did $\Phi$ = $\oint g\bullet dA$ = -4*pi*G*M g$\oint dA$ = -4*pi*GM g[4*pi*r^2] = -4*pi*GM g[4*pi*(R/2)^2] = -4*pi*GM g*pi*R^2 = -4*pi*GM g = (-4GM)/R^2 and since r=R/2 the mass is halved also. therefore g = (-2*G*M)/R^2 i plugged in the values for G, M, and R .. and got -17.40267737 m/s^2 but its not right
 P: 343 anybody can give any hints of what im doing wrong?
Mentor
P: 41,454
 Quote by arl146 and since r=R/2 the mass is halved also.

(What percentage of the sphere's volume--and thus mass, assuming uniform density--is located at r < R/2?)

 P: 343 Gauss' Law for Gravitation ummm .. is the mass 1/8 of M? since V= (4/3)*pi*r^3 and since R=r/2 ... that makes it V = (4/3)*pi*(R^3/8) meaning the volume is 1/8 of the total. and since D = M/V ---> M=DV so the mass also is 1/8 of the original?
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P: 7,808
 Quote by arl146 ummm .. is the mass 1/8 of M? since V= (4/3)*pi*r^3 and since R=r/2 ... that makes it V = (4/3)*pi*(R^3/8) meaning the volume is 1/8 of the total. and since D = M/V ---> M=DV so the mass also is 1/8 of the original?
Assuming that the density of the planet is uniform, yes, that's correct.
 P: 343 is g supposed to be negative? also i got 4.3506693 m/s^2 is that right can someone check for me?
Mentor
P: 41,454
 Quote by arl146 is g supposed to be negative? also i got 4.3506693 m/s^2 is that right can someone check for me?
That looks good. g is negative just means that the field points toward the center.
 P: 343 ok i got it that makes sense. thanks!

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