Q5.1 Calculating the centre of mass I (S)
Calculate the distance of the centre of mass of the earth-moon system, from the centre of the earth. [Mass of earth, me=5.98×1024 kg; mass of moon, mm=7.36×1022 kg; separation, d=3.82×108 m]Hint
Reveal
Solution
Reveal
Hide

![\[ \vec{r}_{cm} = \frac{1}{M}\left[m_1 \vec{r}_1 + m_2 \vec{r}_2 + m_3 \vec{r}_3 + ...\, \right] = \frac{1}{M} \sum_i m_i \vec{r}_i \]](mastermathpng-0.png)

Solution
Let’s choose the centre of the earth as the origin of our (one dimensional) coordinate system, and measure all distances along the vector that joins the earth and moon. The earth is then at the origin, the moon vector has magnitude d.
Now the general definition of the centre of mass (Key Point 4.1) is
![\[ \vec{r}_{cm} = \frac{1}{M}\left[m_1 \vec{r}_1 + m_2 \vec{r}_2 + m_3 \vec{r}_3 + ...\, \right] = \frac{1}{M} \sum_i m_i \vec{r}_i \]](mastermathpng-0.png)
Hence for our 1D system
![\[ x_{cm} = \frac{m_m d}{m_e + m_m} = 4.6 \times 10^6\,m \]](mastermathpng-1.png)