### Finding the Center of MassThe 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 \( x \)-directionThe \( x \)-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 Calculation1. **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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Description: ### Finding the Center of MassThe 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 =

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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=?

Physics for Scientists and Engineers: Foundations and Connections

1st Edition

ISBN:9781133939146

Author:Katz, Debora M.

Publisher:Katz, Debora M.

Chapter10: Systems Of Particles And Conservation Of Momentum

Section10.3: Center Of Mass Revisited

Problem 10.2CE

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Question

### 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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