Answer:
If the mass of the Earth is increased by a factor of 2, then the Fgrav is increased by a factor of 2.
If the mass of the earth is increased by a factor of 3, then Fgrav is increased by a factor of 3.
If the mass of the earth is decreased by a factor of 4, then the Fgrav is decreased by a factor of 4
Explanation:
In order to solve this question, we must take into account that the force of gravity is given by the following formula:
[tex]F_{g0}=G \frac{mM_{E0}}{r^{2}}[/tex]
So if the mass of the earth is increased by a factor of 2, this means that:
[tex]M_{Ef}=2M_{E0}[/tex]
so:
[tex]F_{gf}=G \frac{2mM_{E0}}{r^{2}}[/tex]
Therefore:
[tex]\frac{F_{gf}}{F_{g0}}=\frac{G \frac{2mM_{E0}}{r^{2}}}{G \frac{mM_{E0}}{r^{2}}}[/tex]
When simplifying we end up with:
[tex]\frac{F_{gf}}{F_{g0}}=2[/tex]
so if the mass of the Earth is increased by a factor of 2, then the Fgrav is increased by a factor of 2.
If the mass of the earth is increased by a factor of 3
So if the mass of the earth is increased by a factor of 2, this means that:
[tex]M_{Ef}=3M_{E0}[/tex]
so:
[tex]F_{gf}=G \frac{3mM_{E0}}{r^{2}}[/tex]
Therefore:
[tex]\frac{F_{gf}}{F_{g0}}=\frac{G \frac{3mM_{E0}}{r^{2}}}{G \frac{mM_{E0}}{r^{2}}}[/tex]
When simplifying we end up with:
[tex]\frac{F_{gf}}{F_{g0}}=3[/tex]
so if the mass of the Earth is increased by a factor of 3, then the Fgrav is increased by a factor of 3.
If the mass of the earth is decreased by a factor of 4
So if the mass of the earth is decreased by a factor of 4, this means that:
[tex]M_{Ef}=\frac{M_{E0}}{4}[/tex]
so:
[tex]F_{gf}=G \frac{mM_{E0}}{4r^{2}}[/tex]
Therefore:
[tex]\frac{F_{gf}}{F_{g0}}=\frac{G \frac{mM_{E0}}{4r^{2}}}{G \frac{mM_{E0}}{r^{2}}}[/tex]
When simplifying we end up with:
[tex]\frac{F_{gf}}{F_{g0}}=\frac{1}{4}[/tex]
so if the mass of the Earth is decreased by a factor of 4, then the Fgrav is decreased by a factor of 4.
