# Determining The Mass Of The Earth

• Bashyboy
In summary, the equation to calculate the orbital period of the Earth-Moon system would be incorrect if the Moon were as massive as a neutron star.
Bashyboy

## Homework Statement

(a) Given that the period of the Moon's orbit about the Earth is 27.32 days and the nearly constant distance between the center of the Earth and the center of the Moon is $3.84 \cdot 10^8 m$, use $T^2 = (\frac{4\pi^2}{GM_E})a^3$ to calculate the mass of the Earth.

(b) Why is the value you calculate a bit too large?

## The Attempt at a Solution

For (a), I solved for $M_E$ and got a value of $6.01 \cdot 10^{24}~kg$, of which I am certain is the correct answer, though not 100%. However, my main concern is with part (b); I am honestly don't know how to answer this question. Could someone possibly help me?

Hint: Your formula does not take the mass of moon into account.

So, this formula treats the moon as if it were a point particle? What if the formula did take into account the mass of the moon; how would the formula look in that case?

The formula treats the moon as a massless point particle.
Perfect spheres and points are the same for gravity, and Earth and moon are very close to perfect spheres, so the size does not matter.

What if the formula did take into account the mass of the moon; how would the formula look in that case?
It should be easy to find that formula.

Would it just be the equation to Newton's law of Universal gravitation?

You can derive the formula from Newton's laws if you like.
It looks very similar to the one you have, just with a modification to account for the mass of the second body.

mfb said:
Hint: Your formula does not take the mass of moon into account.

Sure it does. The mass of the moon cancels out from both sides of the force balance equation.

Chestermiller said:
Sure it does. The mass of the moon cancels out from both sides of the force balance equation.
Only with the assumption that the Earth does not move, or with a correction for the distance of moon. In both cases, you have to take the mass of moon into account, or neglect it, which is exactly the source of error which is interesting here.

Chestermiller said:
Sure it does. The mass of the moon cancels out from both sides of the force balance equation.
Both sides of what equation?

Imagine if the Moon was as massive as a neutron star. If this were the case, calculating the orbital period of the Earth-Moon system using just the mass of the Earth will obviously yield the wrong answer. Now it would be much better to just use the mass of the Moon.

In general, the masses of both objects come into play in determining the orbital period. You can ignore the mass of the smaller object only if it is much, much less massive than the larger object. Since the ratio of the Moon's mass to that of the Earth is 0.0123, ignoring the mass of the Moon will yield an answer that is observably too [strike]small[/strike] long.

Last edited:
Very interesting. Thanks for the insight.

## 1. How is the mass of the Earth determined?

The mass of the Earth is determined using Newton's law of gravitation, which states that the force of gravity between two objects is directly proportional to their masses and inversely proportional to the square of the distance between them. By measuring the gravitational force between the Earth and a known mass, such as a satellite, the mass of the Earth can be calculated.

## 2. What is the current accepted value for the mass of the Earth?

The currently accepted value for the mass of the Earth is approximately 5.97 x 10^24 kilograms. This value is constantly being refined and updated as technology and measurement techniques improve.

## 3. How accurate is the measurement of the Earth's mass?

The measurement of the Earth's mass is considered to be very accurate, with a margin of error of only a few percentage points. This is due to the extensive research and advanced technology used in calculating and measuring the Earth's mass.

## 4. Why is it important to determine the mass of the Earth?

Determining the mass of the Earth is important for a variety of reasons. It helps us understand the Earth's gravitational pull and its effects on other objects in our solar system. It also provides valuable information for studying the Earth's structure and composition, as well as for making calculations in fields such as astronomy, geology, and physics.

## 5. Has the mass of the Earth changed over time?

The mass of the Earth is constantly changing due to various factors such as erosion, tectonic activity, and the addition of meteorites. However, these changes are relatively small and do not significantly affect the overall mass of the Earth. The Earth's mass has remained relatively stable over millions of years.

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