Answer :
To find the coordinates of the center of mass of two objects with masses \( m_1 \) and \( m_2 \) located at coordinates \((x_1, y_1)\) and \((x_2, y_2)\) respectively, we use the following formulas:
[tex]\[ x_{\text{center mass}} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \][/tex]
[tex]\[ y_{\text{center mass}} = \frac{m_1 y_1 + m_2 y_2}{m_1 + m_2} \][/tex]
These formulas ensure that the center of mass is the weighted average of the coordinates of the two masses, where the weights are the masses themselves.
Given these values:
- \( m_1 = 5 \)
- \( m_2 = 10 \)
- \( x_1 = 2 \)
- \( y_1 = 3 \)
- \( x_2 = 8 \)
- \( y_2 = 7 \)
Let's calculate the coordinates of the center of mass step-by-step:
1. Calculate the x-coordinate of the center of mass:
[tex]\[ x_{\text{center mass}} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} = \frac{(5 \cdot 2) + (10 \cdot 8)}{5 + 10} = \frac{10 + 80}{15} = \frac{90}{15} = 6.0 \][/tex]
2. Calculate the y-coordinate of the center of mass:
[tex]\[ y_{\text{center mass}} = \frac{m_1 y_1 + m_2 y_2}{m_1 + m_2} = \frac{(5 \cdot 3) + (10 \cdot 7)}{5 + 10} = \frac{15 + 70}{15} = \frac{85}{15} \approx 5.666666666666667 \][/tex]
Thus, the coordinates of the center of mass are:
[tex]\[ (x_{\text{center mass}}, y_{\text{center mass}}) = (6.0, 5.666666666666667) \][/tex]
Now, let's compare these results with the answer choices:
A. \(\left(\frac{m_1 x_1+m_2 x_2}{2\left(m_1+m_2\right)}, \frac{m_1 y_1+m_2 y_2}{2\left(m_1+m_2\right)}\right)\)
B. \(\left(\frac{m_1\left(z_1+z_2\right)}{\left.2 / m_1+m_2\right)}, \frac{m_2\left(y_1+y_2\right)}{2\left(m_1+m_2\right)}\right)\)
C. \(\left(\frac{m_1 z_1+m_1 z_2}{m_1+m_2}, \frac{m_2 y_1+m_1 w_2}{m_1+m_3}\right)\)
D. \(\left(\frac{m_1 a_1+m_1 z_2}{m_1+m_2}, \frac{m_1 y_1+m_2 y_2}{m_1+m_2}\right)\)
Considering our calculated coordinates, the most appropriate option is:
None of the given options exactly match the correct formulas for the center of mass. So, it's important to use the correct formulas for calculating the center of mass as outlined in our step-by-step solution:
[tex]\[ (x_{\text{center mass}}, y_{\text{center mass}}) = \left( \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2}, \frac{m_1 y_1 + m_2 y_2}{m_1 + m_2} \right) \][/tex]
[tex]\[ x_{\text{center mass}} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \][/tex]
[tex]\[ y_{\text{center mass}} = \frac{m_1 y_1 + m_2 y_2}{m_1 + m_2} \][/tex]
These formulas ensure that the center of mass is the weighted average of the coordinates of the two masses, where the weights are the masses themselves.
Given these values:
- \( m_1 = 5 \)
- \( m_2 = 10 \)
- \( x_1 = 2 \)
- \( y_1 = 3 \)
- \( x_2 = 8 \)
- \( y_2 = 7 \)
Let's calculate the coordinates of the center of mass step-by-step:
1. Calculate the x-coordinate of the center of mass:
[tex]\[ x_{\text{center mass}} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} = \frac{(5 \cdot 2) + (10 \cdot 8)}{5 + 10} = \frac{10 + 80}{15} = \frac{90}{15} = 6.0 \][/tex]
2. Calculate the y-coordinate of the center of mass:
[tex]\[ y_{\text{center mass}} = \frac{m_1 y_1 + m_2 y_2}{m_1 + m_2} = \frac{(5 \cdot 3) + (10 \cdot 7)}{5 + 10} = \frac{15 + 70}{15} = \frac{85}{15} \approx 5.666666666666667 \][/tex]
Thus, the coordinates of the center of mass are:
[tex]\[ (x_{\text{center mass}}, y_{\text{center mass}}) = (6.0, 5.666666666666667) \][/tex]
Now, let's compare these results with the answer choices:
A. \(\left(\frac{m_1 x_1+m_2 x_2}{2\left(m_1+m_2\right)}, \frac{m_1 y_1+m_2 y_2}{2\left(m_1+m_2\right)}\right)\)
B. \(\left(\frac{m_1\left(z_1+z_2\right)}{\left.2 / m_1+m_2\right)}, \frac{m_2\left(y_1+y_2\right)}{2\left(m_1+m_2\right)}\right)\)
C. \(\left(\frac{m_1 z_1+m_1 z_2}{m_1+m_2}, \frac{m_2 y_1+m_1 w_2}{m_1+m_3}\right)\)
D. \(\left(\frac{m_1 a_1+m_1 z_2}{m_1+m_2}, \frac{m_1 y_1+m_2 y_2}{m_1+m_2}\right)\)
Considering our calculated coordinates, the most appropriate option is:
None of the given options exactly match the correct formulas for the center of mass. So, it's important to use the correct formulas for calculating the center of mass as outlined in our step-by-step solution:
[tex]\[ (x_{\text{center mass}}, y_{\text{center mass}}) = \left( \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2}, \frac{m_1 y_1 + m_2 y_2}{m_1 + m_2} \right) \][/tex]
