[4.] Two point masses m and 10 m lie along the x-axis, with m at the origin and 10 m at x = L. A third point mass M is moved along the x-axis. (a) At what point is the net gravitational force on M due to the other two masses equal to zero? (b) Sketch the x-component of the net force on M due to m and 10m taking quantities to the right as positive. Include the regions x < 0; 0 < x < L, and x > L. Be especially careful to show the behavior of the graph on either side of x = 0 and x = L.
[4.] Two point masses m and 10 m lie along the x-axis, with m at the origin and 10 m at x = L. A third point mass M is moved along the x-axis. (a) At what point is the net gravitational force on M due to the other two masses equal to zero? (b) Sketch the x-component of the net force on M due to m and 10m taking quantities to the right as positive. Include the regions x < 0; 0 < x < L, and x > L. Be especially careful to show the behavior of the graph on either side of x = 0 and x = L.
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Chapter1: Units, Trigonometry. And Vectors
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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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![### Problem Statement
**4.** Two point masses \( m \) and \( 10m \) lie along the x-axis, with \( m \) at the origin and \( 10m \) at \( x = L \). A third point mass \( M \) is moved along the x-axis.
**(a)** At what point is the net gravitational force on \( M \) due to the other two masses equal to zero?
**(b)** Sketch the x-component of the net force on \( M \) due to \( m \) and \( 10m \), taking quantities to the right as positive. Include the regions \( x < 0 \); \( 0 < x < L \), and \( x > L \). Be especially careful to show the behavior of the graph on either side of \( x = 0 \) and \( x = L \).
---
### Explanation and Solution
The problem involves understanding the gravitational forces between three masses aligned on the x-axis and determining where the forces balance each other out.
#### Part (a) - Equilibrium Point
To find the point where the net gravitational force on \( M \) due to \( m \) and \( 10m \) is zero:
1. **Gravitational Force**
- The gravitational force \( F \) between two masses \( m_1 \) and \( m_2 \) separated by a distance \( r \) is given by:
\[
F = G \frac{m_1 m_2}{r^2}
\]
where \( G \) is the gravitational constant.
2. **Setup the Forces**
- Let the distance from \( m \) at the origin to \( M \) be \( x \).
- The distance from \( 10m \) at \( x = L \) to \( M \) is \( L - x \).
3. **Equating Forces**
- The force on \( M \) due to \( m \) at the origin is:
\[
F_{m \to M} = G \frac{m M}{x^2}
\]
- The force on \( M \) due to \( 10m \) at \( x = L \) is:
\[
F_{10m \to M} = G \frac{10m M}{(L - x](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba4cba5c-1d49-41b8-aac4-ce3ae0c647d3%2Fb8a635a2-6e0f-45d1-bd58-2ffea6fe39ae%2Fh34rrc5_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**4.** Two point masses \( m \) and \( 10m \) lie along the x-axis, with \( m \) at the origin and \( 10m \) at \( x = L \). A third point mass \( M \) is moved along the x-axis.
**(a)** At what point is the net gravitational force on \( M \) due to the other two masses equal to zero?
**(b)** Sketch the x-component of the net force on \( M \) due to \( m \) and \( 10m \), taking quantities to the right as positive. Include the regions \( x < 0 \); \( 0 < x < L \), and \( x > L \). Be especially careful to show the behavior of the graph on either side of \( x = 0 \) and \( x = L \).
---
### Explanation and Solution
The problem involves understanding the gravitational forces between three masses aligned on the x-axis and determining where the forces balance each other out.
#### Part (a) - Equilibrium Point
To find the point where the net gravitational force on \( M \) due to \( m \) and \( 10m \) is zero:
1. **Gravitational Force**
- The gravitational force \( F \) between two masses \( m_1 \) and \( m_2 \) separated by a distance \( r \) is given by:
\[
F = G \frac{m_1 m_2}{r^2}
\]
where \( G \) is the gravitational constant.
2. **Setup the Forces**
- Let the distance from \( m \) at the origin to \( M \) be \( x \).
- The distance from \( 10m \) at \( x = L \) to \( M \) is \( L - x \).
3. **Equating Forces**
- The force on \( M \) due to \( m \) at the origin is:
\[
F_{m \to M} = G \frac{m M}{x^2}
\]
- The force on \( M \) due to \( 10m \) at \( x = L \) is:
\[
F_{10m \to M} = G \frac{10m M}{(L - x
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