the radius of a Planet A is half the radius of planet B. If the mass of A is Ma, what must be the mass of B so that the value of g on b is half that of its value on A??...
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Answers
rA=rB2rA=rB2 (1)
gB=gA2gB=gA2 (2)
From the laws of Physics:
gA=GmArA2gA=GmArA2 (3)
gB=GmBrB2gB=GmBrB2 (4)
Replacing (3) and (4) in (2):
GmBrB2=GmA2rA2GmBrB2=GmA2rA2
mBrB2=mA2rA2mBrB2=mA2rA2 (5)
Replacing (1) in (5):
mBrB2=mA2(rB2)2mBrB2=mA2(rB2)2
mB=2mA
The radius of planet A is half the radius of planet B. If the mass of planet A is Ma, what must be the mass of B? so that the value of g on B is half that of its value on A.
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Originally Answered: The radius of planet A is half the radius of planet B . If the mass of A is Ma, what must be the mass of B so that the value of g on B is half that of its value on A?
If the mass distribution of an object is spherically symmetric, we can assume that its gravitational field outside the surface corresponds to that of a point mass in the center.
The formula for the gravitational acceleration, or specific gravity, g, is
g=GMR2,
so if we want gB=12gA we can substitute
GMBRB2=GMA2RA2
The G's then cancel out, and since RA=12RB:
MBR2B=MA2(RB/2)2
meaning
MB=2MA.
So planet B needs double the mass of planet A in order to have half its gravitational acceleration, being twice as big. This is because g is proportional to the mass, but inversely proportional to the radius, squared.