Two uniform spheres are positioned as shown. Determine the gravitational force F which the titanium sphere exerts on the copper sphere. The value of R is 40 mm. Assume a -5.0,b=2.7, 0-38° 1 R aR Titanium Ө Copper bR ---x
Two uniform spheres are positioned as shown. Determine the gravitational force F which the titanium sphere exerts on the copper sphere. The value of R is 40 mm. Assume a -5.0,b=2.7, 0-38° 1 R aR Titanium Ө Copper bR ---x
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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![**Determining Gravitational Force Between Two Spheres**
**Problem Statement:**
Two uniform spheres are positioned as shown in the accompanying diagram. Determine the gravitational force \( F \) which the titanium sphere exerts on the copper sphere. The value of \( R \) is 40 mm.
Assume:
- \( a = 5.0 \)
- \( b = 2.7 \)
- \( \theta = 38^\circ \)
**Diagram Description:**
The diagram illustrates the positioning of two spheres:
1. A titanium sphere, labeled as "Titanium," with a radius \( R \).
2. A copper sphere, labeled as "Copper," with a radius \( bR \).
Positional Coordinates:
- The titanium sphere is positioned such that the center lies at the point \( (0, R) \) on a Cartesian coordinate system.
- The copper sphere is positioned at \( (aR, bR) \).
The distance between the centers of the spheres is given by a vector defined by components derived from \( aR \) along the \( x \)-axis and \( bR \) along the \( y \)-axis. The angle \( \theta \) between the line connecting the centers of the two spheres and the horizontal axis is specified as \( 38^\circ \).
**Objective:**
Calculate the gravitational force \( F \) exerted by the titanium sphere on the copper sphere using the provided constants \( a \), \( b \), and \( \theta \).
### Important Concepts:
- **Gravitational Force**: The gravitational force between two masses \( m_1 \) and \( m_2 \) is given by the formula:
\[ F = G \frac{m_1 m_2}{r^2} \]
where:
- \( G \) is the gravitational constant,
- \( r \) is the distance between the centers of the two masses.
### Calculation Steps:
1. **Determine the Distance \( r \)**:
\[
r = \sqrt{(aR)^2 + (bR)^2}
\]
2. **Substitute the Known Values**:
\( a = 5.0 \), \( b = 2.7 \), \( R = 40 \text{ mm} \)
3. **Calculate the Gravitational Force**:
After determining \( r \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd5ccfa63-45b7-473c-bb11-56af9dc664c7%2Fbc875295-e82a-4481-807b-92ba9a4857f4%2F2g79i76_processed.png&w=3840&q=75)
Transcribed Image Text:**Determining Gravitational Force Between Two Spheres**
**Problem Statement:**
Two uniform spheres are positioned as shown in the accompanying diagram. Determine the gravitational force \( F \) which the titanium sphere exerts on the copper sphere. The value of \( R \) is 40 mm.
Assume:
- \( a = 5.0 \)
- \( b = 2.7 \)
- \( \theta = 38^\circ \)
**Diagram Description:**
The diagram illustrates the positioning of two spheres:
1. A titanium sphere, labeled as "Titanium," with a radius \( R \).
2. A copper sphere, labeled as "Copper," with a radius \( bR \).
Positional Coordinates:
- The titanium sphere is positioned such that the center lies at the point \( (0, R) \) on a Cartesian coordinate system.
- The copper sphere is positioned at \( (aR, bR) \).
The distance between the centers of the spheres is given by a vector defined by components derived from \( aR \) along the \( x \)-axis and \( bR \) along the \( y \)-axis. The angle \( \theta \) between the line connecting the centers of the two spheres and the horizontal axis is specified as \( 38^\circ \).
**Objective:**
Calculate the gravitational force \( F \) exerted by the titanium sphere on the copper sphere using the provided constants \( a \), \( b \), and \( \theta \).
### Important Concepts:
- **Gravitational Force**: The gravitational force between two masses \( m_1 \) and \( m_2 \) is given by the formula:
\[ F = G \frac{m_1 m_2}{r^2} \]
where:
- \( G \) is the gravitational constant,
- \( r \) is the distance between the centers of the two masses.
### Calculation Steps:
1. **Determine the Distance \( r \)**:
\[
r = \sqrt{(aR)^2 + (bR)^2}
\]
2. **Substitute the Known Values**:
\( a = 5.0 \), \( b = 2.7 \), \( R = 40 \text{ mm} \)
3. **Calculate the Gravitational Force**:
After determining \( r \
![**Calculate the mass of each sphere.**
Assume \( a = 5.0 \), \( b = 2.7 \), \( \theta = 38^\circ \).
**Diagram Explanation:**
The diagram shows two spheres: a Titanium sphere and a Copper sphere. The center of the Titanium sphere is at the origin, marked as \( R \), with an angle \( \theta \) to the horizontal (x-axis).
- The Copper sphere has a radius \( bR \) and its center is at a distance of \( bR \) from the origin at an angle of \( \theta \) degrees from the x-axis.
- The Titanium sphere has a radius \( aR \) and its center is at a distance of \( aR \) from the origin along the y-axis.
**Formulas:**
To find the mass of each sphere, you typically need the volume and density. However, since densities are not provided, the problem might be simplified by their ratios or other factors given.
**Answers:**
- **Titanium sphere** \( m_T = \) [input box] kg
- **Copper sphere** \( m_C = \) [input box] kg](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd5ccfa63-45b7-473c-bb11-56af9dc664c7%2Fbc875295-e82a-4481-807b-92ba9a4857f4%2Fh03kmta_processed.png&w=3840&q=75)
Transcribed Image Text:**Calculate the mass of each sphere.**
Assume \( a = 5.0 \), \( b = 2.7 \), \( \theta = 38^\circ \).
**Diagram Explanation:**
The diagram shows two spheres: a Titanium sphere and a Copper sphere. The center of the Titanium sphere is at the origin, marked as \( R \), with an angle \( \theta \) to the horizontal (x-axis).
- The Copper sphere has a radius \( bR \) and its center is at a distance of \( bR \) from the origin at an angle of \( \theta \) degrees from the x-axis.
- The Titanium sphere has a radius \( aR \) and its center is at a distance of \( aR \) from the origin along the y-axis.
**Formulas:**
To find the mass of each sphere, you typically need the volume and density. However, since densities are not provided, the problem might be simplified by their ratios or other factors given.
**Answers:**
- **Titanium sphere** \( m_T = \) [input box] kg
- **Copper sphere** \( m_C = \) [input box] kg
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