A circular wire hoop of constant density 8=3 lies along the circle x² + y² = 7a² in the xy-plane. Find the hoop's moment of inertia, I2, about the z-axis. The hoop's moment of inertia about the z-axis is l₂ = (Type an exact answer, using as needed.)
A circular wire hoop of constant density 8=3 lies along the circle x² + y² = 7a² in the xy-plane. Find the hoop's moment of inertia, I2, about the z-axis. The hoop's moment of inertia about the z-axis is l₂ = (Type an exact answer, using as needed.)
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section: Chapter Questions
Problem 39RE
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![### Problem: Finding Moment of Inertia of a Circular Wire Hoop
A circular wire hoop of constant density \(\delta = 3\) lies along the circle \(x^2 + y^2 = 7a^2\) in the \(xy\)-plane. Find the hoop's moment of inertia, \(I_z\), about the \(z\)-axis.
The hoop's moment of inertia about the \(z\)-axis is:
\[ I_z = \boxed{\phantom{answer}}. \]
(Type an exact answer, using \(\pi\) as needed.)
### Explanation:
The problem involves calculating the moment of inertia of a circular hoop, which is a common problem in physics and engineering. The density of the wire is given as \(\delta = 3\), and the hoop lies along the circle described by the equation \(x^2 + y^2 = 7a^2\).
To summarize:
1. Identify the given parameters:
- Density \(\delta = 3\)
- Circle equation \(x^2 + y^2 = 7a^2\)
2. Use the formula for the moment of inertia of a circular hoop about the \(z\)-axis:
\[ I_z = \delta \times \text{(Integral over the hoop's circumference)} \]
Ensure to type the exact answer using \(\pi\) as needed in the provided box.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffd49838c-9fe9-4c2b-b877-1f4ac641c60c%2Fe9d82830-1ef3-4d5c-8fc4-1f778a128c88%2Fycijzkw_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem: Finding Moment of Inertia of a Circular Wire Hoop
A circular wire hoop of constant density \(\delta = 3\) lies along the circle \(x^2 + y^2 = 7a^2\) in the \(xy\)-plane. Find the hoop's moment of inertia, \(I_z\), about the \(z\)-axis.
The hoop's moment of inertia about the \(z\)-axis is:
\[ I_z = \boxed{\phantom{answer}}. \]
(Type an exact answer, using \(\pi\) as needed.)
### Explanation:
The problem involves calculating the moment of inertia of a circular hoop, which is a common problem in physics and engineering. The density of the wire is given as \(\delta = 3\), and the hoop lies along the circle described by the equation \(x^2 + y^2 = 7a^2\).
To summarize:
1. Identify the given parameters:
- Density \(\delta = 3\)
- Circle equation \(x^2 + y^2 = 7a^2\)
2. Use the formula for the moment of inertia of a circular hoop about the \(z\)-axis:
\[ I_z = \delta \times \text{(Integral over the hoop's circumference)} \]
Ensure to type the exact answer using \(\pi\) as needed in the provided box.
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