## Step 1: **List Coordinates and Masses** Let's extract the masses and their coordinates from the figure: | Mass | \( m \) | \( x \) | \( y \) | |------|--------|--------|--------| | Blue | 10 kg | 1 | 2 | | Orange | 2 kg | 1 | 4 | | Red | 5 kg | 3 | 4 | | Purple | 7 kg | 6 | 6 | ### Explanation - Each colored dot represents a mass at a certain coordinate \((x, y)\). - We'll use these values to calculate the center of mass. --- ## Step 2: **Calculate the Center of Mass** The center of mass \((x_{cm}, y_{cm})\) is given by: \[ x_{cm} = \frac{\sum m_i x_i}{\sum m_i} \] \[ y_{cm} = \frac{\sum m_i y_i}{\sum m_i} \] Let's calculate: - **Total mass**: \[ M = 10 + 2 + 5 + 7 = 24 \text{ kg} \] - **\(x_{cm}\) calculation**: \[ x_{cm} = \frac{10 \times 1 + 2 \times 1 + 5 \times 3 + 7 \times 6}{24} = \frac{10 + 2 + 15 + 42}{24} = \frac{69}{24} \approx 2.88 \] - **\(y_{cm}\) calculation**: \[ y_{cm} = \frac{10 \times 2 + 2 \times 4 + 5 \times 4 + 7 \times 6}{24} = \frac{20 + 8 + 20 + 42}{24} = \frac{90}{24} = 3.75 \] ### Explanation - We multiply each mass by its \(x\) and \(y\) coordinates, sum them up, and divide by the total mass to find the center of mass. --- ## **Final Answer** **The center of mass is located at:** \[ \boxed{(x_{cm}, y_{cm}) = (2.88, 3.75)} \] This point is the average location of the system's mass.

# Center of Mass Calculation ## Step 1: **List Coordinates and Masses** We extract the masses and their coordinates represented in the figure: | Color | Mass \( m \) | \( x \) | \( y \) | |---------|--------------|---------|---------| | Blue | 10 kg | 1 | 2 | | Orange | 2 kg | 1 | 4 | | Red | 5 kg | 3 | 4 | | Purple | 7 kg | 6 | 6 | ### Explanation - Each color indicates a mass located at coordinates \((x, y)\). - These values will be used to compute the center of mass for the system. --- ## Step 2: **Calculate the Center of Mass** The formula for calculating the center of mass \((x_{cm}, y_{cm})\) is: \[ x_{cm} = \frac{\sum m_i x_i}{\sum m_i} \] \[ y_{cm} = \frac{\sum m_i y_i}{\sum m_i} \] ### Calculations 1. **Total mass**: \[ M = 10 + 2 + 5 + 7 = 24 \text{ kg} \] 2. **Calculating \(x_{cm}\)**: \[ x_{cm} = \frac{10 \times 1 + 2 \times 1 + 5 \times 3 + 7 \times 6}{24} = \frac{10 + 2 + 15 + 42}{24} = \frac{69}{24} \approx 2.88 \] 3. **Calculating \(y_{cm}\)**: \[ y_{cm} = \frac{10 \times 2 + 2 \times 4 + 5 \times 4 + 7 \times 6}{24} = \frac{20 + 8 + 20 + 42}{24} = \frac{90}{24} = 3.75 \] ### Explanation - We compute the weighted average of the coordinates by multiplying each mass by its respective coordinates, summing these products, and dividing by the total mass. --- ## **Final Answer** **The center of mass is located at:** \[ \boxed{(x_{cm}, y_{cm}) = (2.88, 3.75)} \] This point represents the average location of the system's mass, indicating where the total mass can be considered to be concentrated.

# Center of Mass Calculation ## Step 1: **List Coordinates and Masses** We will extract the masses and their coordinates represented in the figure: | Color | Mass \( m \) | \( x \) | \( y \) | |---------|--------------|---------|---------| | Blue | 10 kg | 1 | 2 | | Orange | 2 kg | 1 | 4 | | Red | 5 kg | 3 | 4 | | Purple | 7 kg | 6 | 6 | ### Explanation - Each color corresponds to a mass located at specific coordinates \((x, y)\). - These values will be used in the calculation of the center of mass for the system. --- ## Step 2: **Calculate the Center of Mass** The formula for calculating the center of mass \((x_{cm}, y_{cm})\) is: \[ x_{cm} = \frac{\sum m_i x_i}{\sum m_i} \] \[ y_{cm} = \frac{\sum m_i y_i}{\sum m_i} \] ### Calculations 1. **Total mass**: \[ M = 10 + 2 + 5 + 7 = 24 \text{ kg} \] 2. **Calculating \(x_{cm}\)**: \[ x_{cm} = \frac{10 \times 1 + 2 \times 1 + 5 \times 3 + 7 \times 6}{24} = \frac{10 + 2 + 15 + 42}{24} = \frac{69}{24} \approx 2.88 \] 3. **Calculating \(y_{cm}\)**: \[ y_{cm} = \frac{10 \times 2 + 2 \times 4 + 5 \times 4 + 7 \times 6}{24} = \frac{20 + 8 + 20 + 42}{24} = \frac{90}{24} = 3.75 \] ### Explanation - We compute the weighted average of the coordinates by multiplying each mass by its respective coordinates, summing these products, and dividing by the total mass. --- ## **Final Answer** **The center of mass is located at:** \[ \boxed{(x_{cm}, y_{cm}) = (2.88, 3.75)} \] This point indicates the average location of the system's mass, effectively representing where the total mass can be considered to be concentrated.