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Gauss' Law for Gravitation 
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#1
May2212, 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 


#2
May2312, 03:51 PM

P: 343

anybody can give any hints of what im doing wrong?



#3
May2312, 05:20 PM

Mentor
P: 41,475

(What percentage of the sphere's volumeand thus mass, assuming uniform densityis located at r < R/2?) 


#4
May2312, 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? 


#5
May2312, 09:39 PM

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#6
May2312, 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?



#7
May2412, 04:13 AM

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P: 41,475




#8
May2412, 07:13 PM

P: 343

ok i got it that makes sense. thanks!



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