Physics 195 Midterm 3 Study Guidance

Q1. Where is the center of mass of three point masses (each of mass m) connected by rigid massless rods as shown in the diagram? (x, y) = ...

Background

Topic: Center of Mass

This question tests your understanding of how to calculate the center of mass for a system of discrete point masses in two dimensions.

Key Terms and Formulas:

• Center of mass ($\text{CM}$): The weighted average position of all the masses in a system.

• For point masses, the center of mass coordinates are:

$x_{\text{CM}} = \frac{\sum m_i x_i}{\sum m_i}$

$y_{\text{CM}} = \frac{\sum m_i y_i}{\sum m_i}$

Step-by-Step Guidance

1. Identify the coordinates of each mass. For this diagram, the masses are at (0,0), (a,0), and (a,a).

2. Write the expressions for $x_{\text{CM}}$ and $y_{\text{CM}}$ using the coordinates and masses.

3. Since all masses are equal ($m$), the denominator $\sum m_i$ is $3m$.

4. Plug in the coordinates for each mass into the formulas for $x_{\text{CM}}$ and $y_{\text{CM}}$.

Try solving on your own before revealing the answer!

Final Answer: $(\frac{2}{3}a, \frac{1}{3}a)$

$x_{\text{CM}} = \frac{m \cdot 0 + m \cdot a + m \cdot a}{3m} = \frac{2a}{3}$

$y_{\text{CM}} = \frac{m \cdot 0 + m \cdot 0 + m \cdot a}{3m} = \frac{a}{3}$

The center of mass is located at $(\frac{2}{3}a, \frac{1}{3}a)$, which matches option A.