https://file.aiquickdraw.com/mathbot/file/17300523833428wopc1es.jpg

solve how we got these answers With the origin defined to be at the position of the 150-g mass, 
x
CM
=
−1.23
cm
 and 
y
CM
=
0.69
cm


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.

Three point masses are placed at the corners of a triangle as shown in the figure. Find the center of mass of the three-mass system with the origin defined at the position of the 150-g mass.

Learn how to calculate the center of mass for a three-mass system arranged in a triangular configuration. This example involves point masses of 100 g, 75 g, and 150 g located at specific distances, applying the center of mass formula for a system of discrete masses. Suitable for students in Grades 10-12.

Math Problem Statement

solve how we got these answers With the origin defined to be at the position of the 150-g mass, x CM

−1.23 cm and y CM

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation