Question

question 13
1 pts
a space mining company finds a valuable mineral inside of earths moon, and start mining it. over time they decrease the mass of the moon and transport the minerals they collect to a factory in the asteroid belt. what will happen to the moons speed around the earth over time as its mass decreases and why?
the moons speed will increase, because there is an inverse relationship between the moons mass and its speed
the moons speed will decrease over time because there is an inverse relationship between the moons mass and its speed
the moons speed around the earth is not dependent on the mass of the moon, so it will not change.
the moons speed will decrease because there is a linear relationship between the moons speed and its mass

Explanation:

Step1: Consider the formula for centripetal force and gravitational force

The centripetal force \(F_c = \frac{mv^{2}}{r}\) (where \(m\) is the mass of the moon, \(v\) is its speed, and \(r\) is the radius of its orbit) and the gravitational force \(F_g=\frac{GMm}{r^{2}}\) (where \(M\) is the mass of the Earth, \(G\) is the gravitational constant). At equilibrium \(F_c = F_g\), so \(\frac{mv^{2}}{r}=\frac{GMm}{r^{2}}\).

Step2: Solve for the speed \(v\)

Cancel out the mass \(m\) of the moon from both sides of the equation \(\frac{mv^{2}}{r}=\frac{GMm}{r^{2}}\). We get \(v = \sqrt{\frac{GM}{r}}\).

Answer:

The moon's speed around the Earth is not dependent on the mass of the moon, so it will not change.