Question

practice
directions: answer the following questions using your knowledge of the universal
law of gravitation (ulg).
part i - conceptual
question

1. scientists are trying to decide if earth or

mars has a larger gravity well. if earth has a
mass of 6*10^24 kg and mars has a mass of
6.39*10^23 kg, which has a larger gravity
well? explain your answer.
answer

Explanation:

Step1: Recall the formula for gravitational force

The gravitational force \(F = G\frac{Mm}{r^{2}}\), where \(G\) is the gravitational constant, \(M\) is the mass of the planet, \(m\) is the mass of the object on the planet, and \(r\) is the radius of the planet. For a given object (\(m\) is constant) on the surface of a planet (assuming \(r\) is comparable for Earth and Mars in this conceptual comparison), the gravitational force (and thus the "gravity" experienced by objects on the surface, \(g=\frac{F}{m}=G\frac{M}{r^{2}}\)) is directly proportional to the mass of the planet (\(M\)) when \(r\) is approximately considered in a relative sense.

Step2: Compare the masses of Earth and Mars

Given \(M_{Earth}=6\times 10^{24}\space kg\) and \(M_{Mars}=6.39\times 10^{23}\space kg\). Since \(6\times 10^{24}>6.39\times 10^{23}\) (because \(6\times 10^{24}=60\times 10^{23}\) and \(60\times 10^{23}>6.39\times 10^{23}\)) and using the relationship \(g = G\frac{M}{r^{2}}\) (assuming \(r\) is not the dominant factor in making a rough comparison of surface - level gravity between the two planets based on mass alone as per the problem's context of just using mass information), Earth has a larger mass.

Answer:

Earth has a larger gravity. Because according to the formula \(g = G\frac{M}{r^{2}}\) (where \(g\) is the acceleration due to gravity, \(G\) is the gravitational constant, \(M\) is the mass of the planet, and \(r\) is the radius of the planet), when comparing Earth (\(M = 6\times 10^{24}\space kg\)) and Mars (\(M=6.39\times 10^{23}\space kg\)) and assuming \(r\) is in a comparable range for a basic mass - based comparison of gravity, the larger mass (\(M\)) of Earth leads to a larger \(g\).