I have the problem attached. I also have the formula that will help from the book. I also added some key ideas to help you and you will find that in the attachment of a page from a book, but on the top left. I wrote what you might possibly need. I just think i have my integrals wrong but I do know that the first integral goes to 2 pi.
I have the problem attached. I also have the formula that will help from the book. I also added some key ideas to help you and you will find that in the attachment of a page from a book, but on the top left. I wrote what you might possibly need. I just think i have my integrals wrong but I do know that the first integral goes to 2 pi.
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I have the problem attached. I also have the formula that will help from the book. I also added some key ideas to help you and you will find that in the attachment of a page from a book, but on the top left. I wrote what you might possibly need. I just think i have my integrals wrong but I do know that the first integral goes to 2 pi.
![### Understanding Radial Motion in Cylindrical Systems
In this section, we explore the forces acting on a cylinder, the motion due to gravity, and the pressure exerted. The content stems from principles of classical mechanics and gravitation.
#### Diagram Explanation
The handwritten diagram is a step-by-step process illustrating how to solve gravitational force problems involving a cylindrical structure.
1. **Integral Calculation**:
- Two integrals show the process of summing gravitational contributions over a region, with limits from \(0\) to \(2\pi\) and from \(0\) to \(R\).
- The expression involves integrating the gravitational effect \(-G \frac{M_r \rho}{r^2}\), indicating contributions at various radial positions.
2. **Gravitational Force \( F_g \) Equation**:
- Shows the integration setup for determining gravitational force:
\( F_G = \iint dV \, \vec{G} \)
where the integrals cover volumes influenced by the gravitational field.
#### Textual Explanation
The textual excerpt explains the forces on a cylinder and derives the equation for radial motion considering spherical symmetry:
1. **Pressure Differential and Force**:
- The differential force \( dF_P = A dP \) (Equation 10.2) represents pressure difference effects across the cylinder surfaces due to external forces.
2. **Equation Integration**:
- Combining equations (10.2) and (10.3) gives:
\[
dm \frac{d^2r}{dt^2} = -G \frac{M_r dm}{r^2} - A dP
\]
- The equation represents the balance between gravitational attraction and pressure differences acting radially on the cylinder.
3. **Density Considerations**:
- The density \( \rho \) of the gas leads to mass expression:
\[
dm = \rho A dr
\]
- This allows substitution into the motion equation.
4. **Radial Motion Equation**:
- Dividing by the cylinder's volume simplifies:
\[
\rho \frac{d^2r}{dt^2} = -G \frac{M_r \rho}{r^2} - \frac{dP}{dr}
\]
- This is the fundamental equation describing radial acceleration, under gravitational influence and pressure gradient.
This set](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6214fda8-f992-4e88-b320-5195339361f5%2F141d90be-00c7-4eea-a1ff-386b32acf63d%2Fuo2lpwa_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Radial Motion in Cylindrical Systems
In this section, we explore the forces acting on a cylinder, the motion due to gravity, and the pressure exerted. The content stems from principles of classical mechanics and gravitation.
#### Diagram Explanation
The handwritten diagram is a step-by-step process illustrating how to solve gravitational force problems involving a cylindrical structure.
1. **Integral Calculation**:
- Two integrals show the process of summing gravitational contributions over a region, with limits from \(0\) to \(2\pi\) and from \(0\) to \(R\).
- The expression involves integrating the gravitational effect \(-G \frac{M_r \rho}{r^2}\), indicating contributions at various radial positions.
2. **Gravitational Force \( F_g \) Equation**:
- Shows the integration setup for determining gravitational force:
\( F_G = \iint dV \, \vec{G} \)
where the integrals cover volumes influenced by the gravitational field.
#### Textual Explanation
The textual excerpt explains the forces on a cylinder and derives the equation for radial motion considering spherical symmetry:
1. **Pressure Differential and Force**:
- The differential force \( dF_P = A dP \) (Equation 10.2) represents pressure difference effects across the cylinder surfaces due to external forces.
2. **Equation Integration**:
- Combining equations (10.2) and (10.3) gives:
\[
dm \frac{d^2r}{dt^2} = -G \frac{M_r dm}{r^2} - A dP
\]
- The equation represents the balance between gravitational attraction and pressure differences acting radially on the cylinder.
3. **Density Considerations**:
- The density \( \rho \) of the gas leads to mass expression:
\[
dm = \rho A dr
\]
- This allows substitution into the motion equation.
4. **Radial Motion Equation**:
- Dividing by the cylinder's volume simplifies:
\[
\rho \frac{d^2r}{dt^2} = -G \frac{M_r \rho}{r^2} - \frac{dP}{dr}
\]
- This is the fundamental equation describing radial acceleration, under gravitational influence and pressure gradient.
This set
Transcribed Image Text:1) Prove that the gravitational force on a point mass \( m \) located in the center of a hollow spherically symmetric shell with mass \( M \) is zero. Assume that the radius of the inside surface of the shell is \( r_1 \) and the radius of the outside surface \( r_2 \). Use \( d\vec{F}_G \) and \( dM \) and integrate in spherical coordinates. Since in this problem the direction of the force is important, use \( \hat{r} = \sin \theta \cos \varphi \, \hat{x} + \sin \theta \sin \varphi \, \hat{y} + \cos \theta \, \hat{z} \), with \( \hat{x}, \hat{y} \) and \( \hat{z} \) independent of \( \theta \) and \( \varphi \). Use \( A = r^2 \sin \theta \, d\theta \, d\varphi \) (blue area in figure).
The diagram shows a hollow spherical shell with radii \( r_1 \) and \( r_2 \). The shell is represented in a spherical coordinate system with axes labeled \( x \), \( y \), and \( z \). A point \( P \) is considered within the shell, and its location is determined using spherical coordinates: the radial distance \( r \), polar angle \( \theta \), and azimuthal angle \( \varphi \). The arc length along the shell's surface is depicted, illustrating the positional relationship within the spherical coordinates. A small surface area element \( dA \) is highlighted in blue, and this corresponds to the expression \( r^2 \sin \theta \, d\theta \, d\varphi \).
Expert Solution
Step 1
Solution:
Given the mass of the spherical shell is M. Let's consider a small mass element dM with surface area A. Equating the area density of small mass dM and the total mass M.
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