Question

The mass of the moon is about $0.013$ times the mass of the earth and the distance from the center of mass to the center of the earth is about $60$ times the radius of the earth. Show that the center of mass of the earth-moon system lies inside the earth and find its approximation from the center of the earth. Take the earth’s radius to be $6400\text{ km}$.

Answer 1

Hint: Center of mass of a system can be defined as the point at which we can concentrate all the mass of the system. To solve these types of questions, we must have knowledge about the formula used to calculate the center of mass of a system using the mass of the system and distance of the bodies in the system from the center of mass.

Complete step-by-step solution: -
Let us assume the mass of earth to be $m\text{ kg}$, then the mass of the moon which is about $0.013$ times the mass of the earth would be $0.013m$. Let us consider that the center of the earth lies at origin. Then the distance of the moon from the center of the earth which is given to be about $60$ times the radius of the earth will be $60R$, where $R$is the radius of the earth.
The center of mass of the earth-moon system can be found out with the following formula:
$CM={\displaystyle \frac{m_1{x}_1+m_2{x}_2}{m_1+m_2}}$
On substituting the values in the formula, we get:
$$\begin{array}{rl}& CM={\displaystyle \frac{m\times 0+0.013m\times 60R}{m+0.013m}}\\ & \Rightarrow CM={\displaystyle \frac{0.78mR}{1.013m}}\\ & \Rightarrow CM={\displaystyle \frac{0.78R}{1.013}}\\ & \Rightarrow CM=0.76999R\end{array}$$
Hence, it is proved that the center of mass of the earth-moon system lies inside the earth.
We know that the radius of earth is given as $6400\text{ km}$, hence on substituting the value in the equation, we get:
$\begin{array}{rl}& CM=0.76999\times 6400\\ & \Rightarrow CM=\text{4927}\text{.936 km}\\ & \therefore CM\approx \text{4928 km}\end{array}$
Thus, the approximate value of the distance of center of mass of the earth-moon system from the center of earth comes out to be $4928\text{ km}$.

Note: We must keep in mind that the center of mass of a system will always lie inside the system. It can be defined as that point which can be assumed to carry all the mass of the system. It must not be confused with center of gravity as these two terms are quite different.