Gauss' Law for Gravitation

arl146

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:

[itex]\Phi[/itex] = [itex]\oint g\bullet dA[/itex] = -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

[itex]\Phi[/itex] = [itex]\oint g\bullet dA[/itex] = -4*pi*G*M
g[itex]\oint dA[/itex] = -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

arl146

anybody can give any hints of what im doing wrong?

Doc Al

Are you sure about that?

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

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?

SammyS

Assuming that the density of the planet is uniform, yes, that's correct.

arl146

is g supposed to be negative? also i got 4.3506693 m/s^2 is that right can someone check for me?

Doc Al

That looks good. g is negative just means that the field points toward the center.

arl146

ok i got it that makes sense. thanks!