College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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![### Rotational Inertia Calculation
In this exercise, we will calculate the rotational inertia of three masses with respect to the y-axis. The position and mass of each body are given below:
- **Mass 1 (m1)**:
- Mass: 3.5 units
- Position: \( x_1 = 4.4 \), \( y_1 = -2.6 \)
- **Mass 2 (m2)**:
- Mass: 5.0 units
- Position: \( x_2 = -4.4 \), \( y_2 = -2.8 \)
- **Mass 3 (m3)**:
- Mass: 3.0 units
- Position: \( x_3 = -3.2 \), \( y_3 = -2.6 \)
### Explanation
Rotational inertia (also known as moment of inertia) with respect to the y-axis, \( I_y \), can be calculated using the formula:
\[ I_y = \sum m_i \cdot x_i^2 \]
where:
- \( m_i \) is the mass of the ith object.
- \( x_i \) is the x-coordinate of the ith object.
Using the provided data, we can calculate the total rotational inertia around the y-axis by summing the contributions of each individual mass.
### Detailed Steps
For each mass, we will compute the product of the mass and the square of its x-coordinate.
1. **Mass 1 (m1 = 3.5)**:
\[ I_{y1} = m1 \cdot x1^2 = 3.5 \cdot (4.4)^2 \]
2. **Mass 2 (m2 = 5.0)**:
\[ I_{y2} = m2 \cdot x2^2 = 5.0 \cdot (-4.4)^2 \]
3. **Mass 3 (m3 = 3.0)**:
\[ I_{y3} = m3 \cdot x3^2 = 3.0 \cdot (-3.2)^2 \]
Summing these values will give the total rotational inertia with respect to the y-axis:
\[ I_y = I_{y1} + I_{y2} + I_{y3} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc0f85ff8-2ea0-4d43-a0c0-84de96cc63b8%2F82e70293-a51e-4159-886c-7eb2c105f1fb%2Fr6m51m_processed.png&w=3840&q=75)
Transcribed Image Text:### Rotational Inertia Calculation
In this exercise, we will calculate the rotational inertia of three masses with respect to the y-axis. The position and mass of each body are given below:
- **Mass 1 (m1)**:
- Mass: 3.5 units
- Position: \( x_1 = 4.4 \), \( y_1 = -2.6 \)
- **Mass 2 (m2)**:
- Mass: 5.0 units
- Position: \( x_2 = -4.4 \), \( y_2 = -2.8 \)
- **Mass 3 (m3)**:
- Mass: 3.0 units
- Position: \( x_3 = -3.2 \), \( y_3 = -2.6 \)
### Explanation
Rotational inertia (also known as moment of inertia) with respect to the y-axis, \( I_y \), can be calculated using the formula:
\[ I_y = \sum m_i \cdot x_i^2 \]
where:
- \( m_i \) is the mass of the ith object.
- \( x_i \) is the x-coordinate of the ith object.
Using the provided data, we can calculate the total rotational inertia around the y-axis by summing the contributions of each individual mass.
### Detailed Steps
For each mass, we will compute the product of the mass and the square of its x-coordinate.
1. **Mass 1 (m1 = 3.5)**:
\[ I_{y1} = m1 \cdot x1^2 = 3.5 \cdot (4.4)^2 \]
2. **Mass 2 (m2 = 5.0)**:
\[ I_{y2} = m2 \cdot x2^2 = 5.0 \cdot (-4.4)^2 \]
3. **Mass 3 (m3 = 3.0)**:
\[ I_{y3} = m3 \cdot x3^2 = 3.0 \cdot (-3.2)^2 \]
Summing these values will give the total rotational inertia with respect to the y-axis:
\[ I_y = I_{y1} + I_{y2} + I_{y3} \]
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