The gravitational field g due to a point mass M may be obtained by analogy with
ID: 1398887 • Letter: T
Question
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 = ointg.dA=-4*pi*GM
where G is the gravitational constant.
Use this result to calculate the gravitational acceleration g at a distance of R/2 from the center of a planet of radius R = 5.60 x 1006 m and M = 7.50 x 1024 kg.
Explanation / Answer
integration (g.dA) = -4*pi*G*M
Which will teurn out to be:
g*A = -4*pi*G*M
Where:
A is area = 4*pi*(R/2)^2
= 4*pi*(5.6*10^6/2)^2
=3.941*10^14 m^2
Total mass = 7.5*10^24 kg
Total volume = (4/3)*pi*(R)^3 = (4/3)*PI*(5.6*10^6)^3 = 7.3562*10^20 m^3
density = total mass/total volume = 7.5*10^24 / (7.3562*10^20)= 10195.48 Kg/m^3
M= mass enclosed by this area = density * volume of this area
= 10195.48 * (4/3 *pi*(R/2)^3)
= 10195.48 * (4/3 *pi*(5.6*10^6/2)^3)
= 9.4*10^23 Kg
Putting all these values in:
g*A = -4*pi*G*M
g*(3.941*10^14) = -4*pi*(6.673*10^-11)*(9.4*10^23)
g = 2 m/s^2
Answer: 2 m/s^2
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