https://file.aiquickdraw.com/mathbot/file/1729018150769kd4em50g.jpg

To solve Part (a) of this problem, we need to find the \( x \)-coordinate of the center of mass (\( x_{cm} \)) for the three-mass system. The formula for the \( x \)-coordinate of the center of mass is:

\[
x_{cm} = \frac{m_1 x\[
x_{cm} = \frac{m_1 x_1 + m_2 x_2 + m_3 x_3}{m_1 + m_2 + m_3}
\]

where:
- \( m_1, m_2, m_3 \) are the masses at the three points,
- \( x_1, x_2, x_3 \) are the \( x \)-coordinates of each mass.

From the problem:
- \( m_1 = 71 \) g, \( m_2 = 51 \) g, \( m_3 = 120 \) g.
- The coordinates given in the diagram are:
  - \( m_1 \) is at the origin: \( (0, 0) \), so \( x_1 = 0 \).
  - \( m_2 \) is at \( (0, w) \), which means \( x_2 = 0 \) (since it lies on the y-axis).
  - \( m_3 \) is at \( (l, 0) \), so \( x_3 = l \).

Given values:
- \( l = 4.01 \) cm, \( w = 3.01 \) cm.

Now we can substitute these values:

\[
x_{cm} = \frac{(71 \cdot 0) + (51 \cdot 0) + (120 \cdot 4.01)}{71 + 51 + 120}
\]

\[
x_{cm} = \frac{0 + 0 + 481.2}{242}
\]

\[
x_{cm} = \frac{481.2}{242} \approx 1.99 \text{ cm}
\]

So, the \( x \)-coordinate of the center of mass is approximately \( 1.99 \) cm.

Would you like a detailed explanation of this process, or do you have any questions?

Here are 5 related questions to further your understanding:
1. How would you find the \( y \)-coordinate of the center of mass for this system?
2. What is the physical significance of the center of mass?
3. How does the mass distribution affect the location of the center of mass?
4. If \( m_2 \) were moved to a different coordinate, how would that change \( x_{cm} \)?
5. How would you find the center of mass in a three-dimensional coordinate system?

**Tip:** Always check if the coordinates of each mass point match their placement on the graph, as incorrect coordinates can lead to errors in calculating the center of mass.

Three point masses are arranged on the corners of a right triangle. The masses are m1 = 71 g, m2 = 51 g, and m3 = 120 g. The side lengths of the triangle are l = 4.01 cm and w = 3.01 cm. Write an expression for the x-coordinate of the center of mass of the three-mass system.

Learn how to calculate the x-coordinate of the center of mass for a system of three point masses arranged on a right triangle. This problem involves applying the center of mass formula with given masses m1 = 71 g, m2 = 51 g, and m3 = 120 g, and side lengths l = 4.01 cm, w = 3.01 cm. Suitable for introductory college-level physics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation