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Gauss' Law for Gravitation

by arl146
Tags: gauss, gravitation
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arl146
#1
May22-12, 11:18 PM
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:

[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
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arl146
#2
May23-12, 03:51 PM
P: 343
anybody can give any hints of what im doing wrong?
Doc Al
#3
May23-12, 05:20 PM
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Quote Quote by arl146 View Post
and since r=R/2 the mass is halved also.
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
#4
May23-12, 09:05 PM
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?
SammyS
#5
May23-12, 09:39 PM
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Quote Quote by arl146 View Post
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.
arl146
#6
May23-12, 10:00 PM
P: 343
is g supposed to be negative? also i got 4.3506693 m/s^2 is that right can someone check for me?
Doc Al
#7
May24-12, 04:13 AM
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Quote Quote by arl146 View Post
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.
arl146
#8
May24-12, 07:13 PM
P: 343
ok i got it that makes sense. thanks!


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