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]
