Find the x center of mass of the following masses at their respective coordinates: m1=2.5, x1=0.6,y1=-0.6 m2=3.5, x2=2.8,y2=3.8 m3=4.0, x3=-2.4,y3=-2.6 X_com=?
Find the x center of mass of the following masses at their respective coordinates: m1=2.5, x1=0.6,y1=-0.6 m2=3.5, x2=2.8,y2=3.8 m3=4.0, x3=-2.4,y3=-2.6 X_com=?
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Chapter1: Units, Trigonometry. And Vectors
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![### Finding the Center of Mass
The task is to find the \( x \)-coordinate of the center of mass for the following set of masses and their respective coordinates:
**Masses and Coordinates:**
- \( m_1 = 2.5 \), \( x_1 = 0.6 \), \( y_1 = -0.6 \)
- \( m_2 = 3.5 \), \( x_2 = 2.8 \), \( y_2 = 3.8 \)
- \( m_3 = 4.0 \), \( x_3 = -2.4 \), \( y_3 = -2.6 \)
**Find:** \( x_{\text{com}} \)
### Formula for Center of Mass in the \( x \)-direction
The \( x \)-coordinate of the center of mass (\( x_{\text{com}} \)) is given by the formula:
\[ x_{\text{com}} = \frac{\sum (m_i \cdot x_i)}{\sum m_i} \]
### Step-by-Step Calculation
1. **Calculate the weighted \( x \)-coordinates (numerator):**
\[
(m_1 \cdot x_1) + (m_2 \cdot x_2) + (m_3 \cdot x_3)
\]
- For \( m_1 = 2.5 \) and \( x_1 = 0.6 \):
\[
2.5 \times 0.6 = 1.5
\]
- For \( m_2 = 3.5 \) and \( x_2 = 2.8 \):
\[
3.5 \times 2.8 = 9.8
\]
- For \( m_3 = 4.0 \) and \( x_3 = -2.4 \):
\[
4.0 \times -2.4 = -9.6
\]
Add the weighted coordinates:
\[
1.5 + 9.8 - 9.6 = 1.7
\]
2. **Calculate the sum of the masses (denominator):**
\[
m_1 + m_2 + m_3 = 2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbda06ec9-3160-4152-b2ee-882d8e600c29%2Fd1fb8d63-cb5e-48f2-9e20-f51a431b3dfe%2Fi3savtxf_processed.png&w=3840&q=75)
Transcribed Image Text:### Finding the Center of Mass
The task is to find the \( x \)-coordinate of the center of mass for the following set of masses and their respective coordinates:
**Masses and Coordinates:**
- \( m_1 = 2.5 \), \( x_1 = 0.6 \), \( y_1 = -0.6 \)
- \( m_2 = 3.5 \), \( x_2 = 2.8 \), \( y_2 = 3.8 \)
- \( m_3 = 4.0 \), \( x_3 = -2.4 \), \( y_3 = -2.6 \)
**Find:** \( x_{\text{com}} \)
### Formula for Center of Mass in the \( x \)-direction
The \( x \)-coordinate of the center of mass (\( x_{\text{com}} \)) is given by the formula:
\[ x_{\text{com}} = \frac{\sum (m_i \cdot x_i)}{\sum m_i} \]
### Step-by-Step Calculation
1. **Calculate the weighted \( x \)-coordinates (numerator):**
\[
(m_1 \cdot x_1) + (m_2 \cdot x_2) + (m_3 \cdot x_3)
\]
- For \( m_1 = 2.5 \) and \( x_1 = 0.6 \):
\[
2.5 \times 0.6 = 1.5
\]
- For \( m_2 = 3.5 \) and \( x_2 = 2.8 \):
\[
3.5 \times 2.8 = 9.8
\]
- For \( m_3 = 4.0 \) and \( x_3 = -2.4 \):
\[
4.0 \times -2.4 = -9.6
\]
Add the weighted coordinates:
\[
1.5 + 9.8 - 9.6 = 1.7
\]
2. **Calculate the sum of the masses (denominator):**
\[
m_1 + m_2 + m_3 = 2
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