The center of mass is defined as a point on the straight line between two objects with masses [tex]$m_1[tex]$[/tex] and [tex]$[/tex]m_2[tex]$[/tex] such that [tex]$[/tex]\frac{L_1}{L_2}=\frac{m_2}{m_1}$[/tex].

What are the coordinates of the center of mass if:

[tex]$m_2 = (x_2, y_2)$[/tex]

[tex]$m_1 = (x_1, y_1)$[/tex]?



Answer :

To find the coordinates of the center of mass for two objects with masses \( m_1 \) and \( m_2 \), located at coordinates \((x_1, y_1)\) and \((x_2, y_2)\) respectively, you can use the following formulas:

[tex]\[ x_{\text{cm}} = \frac{m_1 \cdot x_1 + m_2 \cdot x_2}{m_1 + m_2} \][/tex]
[tex]\[ y_{\text{cm}} = \frac{m_1 \cdot y_1 + m_2 \cdot y_2}{m_1 + m_2} \][/tex]

Let's solve the problem with the given data:

- Mass of the first object \( m_1 = 5 \)
- Coordinates of the first object \((x_1, y_1) = (2, 3)\)
- Mass of the second object \( m_2 = 10 \)
- Coordinates of the second object \((x_2, y_2) = (8, 6)\)

First, let's find the x-coordinate of the center of mass (\( x_{\text{cm}} \)):

[tex]\[ x_{\text{cm}} = \frac{(5 \cdot 2) + (10 \cdot 8)}{5 + 10} = \frac{10 + 80}{15} = \frac{90}{15} = 6.0 \][/tex]

Next, we find the y-coordinate of the center of mass (\( y_{\text{cm}} \)):

[tex]\[ y_{\text{cm}} = \frac{(5 \cdot 3) + (10 \cdot 6)}{5 + 10} = \frac{15 + 60}{15} = \frac{75}{15} = 5.0 \][/tex]

Therefore, the coordinates of the center of mass are:

[tex]\[ \boxed{(6.0, 5.0)} \][/tex]