To find the center of mass (CM) of a system of point masses, we need to apply the following formula for both the $x$- and $y$-coordinates.

Formulas for center of mass:

$x_{CM} = \frac{\sum m_i x_i}{\sum m_i}, \quad y_{CM} = \frac{\sum m_i y_i}{\sum m_i}$

where:

• $m_i$ is the mass at the $i^{th}$ point.
• $x_i$ and $y_i$ are the coordinates of the $i^{th}$ mass.
• $\sum m_i$ is the total mass of the system.

Step 1: Assigning Coordinates

The problem states that the origin is defined at the position of the 150-g mass. Therefore, its coordinates are:

• Mass 1: $m_1 = 150 \, g$, $x_1 = 0 \, cm$, $y_1 = 0 \, cm$

The 100-g mass is located 4 cm to the left of the origin along the x-axis. Its coordinates are:

• Mass 2: $m_2 = 100 \, g$, $x_2 = -4 \, cm$, $y_2 = 0 \, cm$

The 75-g mass is located 3 cm above the 150-g mass along the y-axis. Its coordinates are:

• Mass 3: $m_3 = 75 \, g$, $x_3 = 0 \, cm$, $y_3 = 3 \, cm$

Step 2: Calculate the Total Mass

$\text{Total Mass} = m_1 + m_2 + m_3 = 150 \, g + 100 \, g + 75 \, g = 325 \, g$

Step 3: Calculate $x_{CM}$

$x_{CM} = \frac{(m_1 \cdot x_1) + (m_2 \cdot x_2) + (m_3 \cdot x_3)}{m_1 + m_2 + m_3}$

$x_{CM} = \frac{(150 \cdot 0) + (100 \cdot -4) + (75 \cdot 0)}{325}$

$x_{CM} = \frac{0 - 400 + 0}{325} = \frac{-400}{325} \approx -1.23 \, cm$

Step 4: Calculate $y_{CM}$

$y_{CM} = \frac{(m_1 \cdot y_1) + (m_2 \cdot y_2) + (m_3 \cdot y_3)}{m_1 + m_2 + m_3}$

$y_{CM} = \frac{(150 \cdot 0) + (100 \cdot 0) + (75 \cdot 3)}{325}$

$y_{CM} = \frac{0 + 0 + 225}{325} = \frac{225}{325} \approx 0.69 \, cm$

Final Answer

The coordinates of the center of mass are:

$x_{CM} = -1.23 \, cm, \quad y_{CM} = 0.69 \, cm$

Do you want more explanation, or is any part unclear?

Related Questions:

1. How does changing the origin affect the calculation of the center of mass?
2. What happens to the center of mass if one of the masses becomes zero?
3. How would the result change if the 75-g mass were shifted horizontally?
4. How can the concept of the center of mass be applied to continuous mass distributions?
5. What is the physical significance of the center of mass in a moving system?

Tip: Always carefully assign coordinates and verify distances when working with the center of mass. Even small coordinate errors can lead to incorrect answers.