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is the acceleration due to gravity of the earth greater than the moons
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Yes because the gravity of the earth is greater than the moon. Why? because the mass of the earth is greater than that of the moon.
It helps to think about how gravitational acceleration is calculated: \[g=\frac{ GM _{p} }{ r ^{2} }\] Where G is the universal gravitational constant (same for all planets, or moons) M is the mass of the planet (or moon) and r is the radius of the planet (or moon) We find that because the Earth is so massive compared to the moon the gravitational acceleration (g) is larger for the Earth.
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