To solve this problem, we need to find the center of mass (CM) of the U-shaped system formed by three uniform thin rods.

Given:

• Length of each rod: $L = 150 \, \text{cm}$
• Mass of each vertical rod: $m_v = 200 \, \text{g}$
• Mass of the horizontal rod: $m_h = 600 \, \text{g}$

Approach:

The center of mass is found by calculating the weighted average of the $x$- and $y$-coordinates of each rod.

1. Horizontal Rod (Mass $m_h$):

• Positioned horizontally along the x-axis, between $x = 0$ and $x = L$.
• The center of mass of this rod lies at $x_h = \frac{L}{2} = 75 \, \text{cm}$.
• Since the rod is along the x-axis, its $y$-coordinate is $y_h = 0 \, \text{cm}$.

2. Left Vertical Rod (Mass $m_v$):

• Positioned along the y-axis at $x = 0$, between $y = 0$ and $y = L$.
• The center of mass of this rod lies at $x_{v1} = 0 \, \text{cm}$, and its $y$-coordinate is $y_{v1} = \frac{L}{2} = 75 \, \text{cm}$.

3. Right Vertical Rod (Mass $m_v$):

• Positioned along the y-axis at $x = L$, between $y = 0$ and $y = L$.
• The center of mass of this rod lies at $x_{v2} = L = 150 \, \text{cm}$, and its $y$-coordinate is $y_{v2} = \frac{L}{2} = 75 \, \text{cm}$.

Center of Mass Calculation:

The general formula for the center of mass is: $x_{\text{cm}} = \frac{\sum m_i x_i}{\sum m_i} \quad \text{and} \quad y_{\text{cm}} = \frac{\sum m_i y_i}{\sum m_i}$ Where $m_i$ is the mass of each rod, and $x_i$ and $y_i$ are the coordinates of their center of mass.

$x_{\text{cm}}$ Calculation:

$x_{\text{cm}} = \frac{m_v \cdot x_{v1} + m_h \cdot x_h + m_v \cdot x_{v2}}{m_v + m_h + m_v}$ Substituting the values: $x_{\text{cm}} = \frac{200 \cdot 0 + 600 \cdot 75 + 200 \cdot 150}{200 + 600 + 200} = \frac{0 + 45000 + 30000}{1000} = \frac{75000}{1000} = 75 \, \text{cm}$

$y_{\text{cm}}$ Calculation:

$y_{\text{cm}} = \frac{m_v \cdot y_{v1} + m_h \cdot y_h + m_v \cdot y_{v2}}{m_v + m_h + m_v}$ Substituting the values: $y_{\text{cm}} = \frac{200 \cdot 75 + 600 \cdot 0 + 200 \cdot 75}{200 + 600 + 200} = \frac{15000 + 0 + 15000}{1000} = \frac{30000}{1000} = 30 \, \text{cm}$

Final Answer:

• The x-coordinate of the center of mass: $x_{\text{cm}} = 75 \, \text{cm}$
• The y-coordinate of the center of mass: $y_{\text{cm}} = 30 \, \text{cm}$

Would you like more details or have any questions?

Follow-up Questions:

1. How would the center of mass change if the horizontal rod's mass increased?
2. What happens to the center of mass if one of the vertical rods is removed?
3. Can you calculate the moment of inertia of this system about the x-axis?
4. How would the center of mass change if the vertical rods were longer?
5. What would happen if the rods were not uniform but had varying mass density?

Tip:

The center of mass depends not just on the position of the objects, but also on their relative masses.