■The Moon's mass is 1/80 that of Earth, and the lunar radius is 1/4 Earth's radius. Based on these figures, calculate the total weight on the Moon of a 100-kg astronaut with a 50-kg space- suit and backpack, relative to his or her weight on Earth.
■The Moon's mass is 1/80 that of Earth, and the lunar radius is 1/4 Earth's radius. Based on these figures, calculate the total weight on the Moon of a 100-kg astronaut with a 50-kg space- suit and backpack, relative to his or her weight on Earth.
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![### Calculation of Weight on the Moon
**Problem**:
The Moon’s mass is 1/80 that of Earth, and the lunar radius is 1/4 Earth’s radius. Based on these figures, calculate the total weight on the Moon of a 100-kg astronaut with a 50-kg spacesuit and backpack, relative to his or her weight on Earth.
#### Step-by-Step Solution:
1. **Understanding the Figures**:
- Mass of the Moon \(M_{\text{Moon}} = \frac{1}{80} \times M_{\text{Earth}}\)
- Radius of the Moon \(R_{\text{Moon}} = \frac{1}{4} \times R_{\text{Earth}}\)
2. **Gravitational Force Formula**:
The weight of an object is determined by the gravitational force, which is given by the formula:
\[
F = \frac{G \times m_1 \times m_2}{r^2}
\]
Here, \(F\) is the gravitational force, \(G\) is the universal gravitational constant, \(m_1\) and \(m_2\) are the masses of the two objects, and \(r\) is the distance between the centers of the two objects (the radius of the planet in this case).
3. **Setting Up the Weight Comparison**:
We need to compare weights on Earth and the Moon. Since \(G\), the astronaut's mass (\(m\)), and the mass of the equipment (\(m_{\text{equipment}}\)) are constants, we can use:
\[
\frac{W_{\text{Moon}}}{W_{\text{Earth}}} = \frac{M_{\text{Moon}}}{M_{\text{Earth}}} \times \left(\frac{R_{\text{Earth}}}{R_{\text{Moon}}}\right)^2
\]
4. **Substitute Known Values**:
\[
W_{\text{Moon}} = W_{\text{Earth}} \times \left(\frac{1/80}{1} \times \left(\frac{4}{1}\right)^2\right)
\]
Simplify the equation:
\[
W_{\text{Moon}} = W_{\text{Earth}} \times \left(\frac{1}{80} \times](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4fdfc26a-e6d8-4dfd-b2bf-654dccb4509d%2Fc8ee1eec-329a-4066-b4ca-1d8968142877%2Fbo7y6jx_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculation of Weight on the Moon
**Problem**:
The Moon’s mass is 1/80 that of Earth, and the lunar radius is 1/4 Earth’s radius. Based on these figures, calculate the total weight on the Moon of a 100-kg astronaut with a 50-kg spacesuit and backpack, relative to his or her weight on Earth.
#### Step-by-Step Solution:
1. **Understanding the Figures**:
- Mass of the Moon \(M_{\text{Moon}} = \frac{1}{80} \times M_{\text{Earth}}\)
- Radius of the Moon \(R_{\text{Moon}} = \frac{1}{4} \times R_{\text{Earth}}\)
2. **Gravitational Force Formula**:
The weight of an object is determined by the gravitational force, which is given by the formula:
\[
F = \frac{G \times m_1 \times m_2}{r^2}
\]
Here, \(F\) is the gravitational force, \(G\) is the universal gravitational constant, \(m_1\) and \(m_2\) are the masses of the two objects, and \(r\) is the distance between the centers of the two objects (the radius of the planet in this case).
3. **Setting Up the Weight Comparison**:
We need to compare weights on Earth and the Moon. Since \(G\), the astronaut's mass (\(m\)), and the mass of the equipment (\(m_{\text{equipment}}\)) are constants, we can use:
\[
\frac{W_{\text{Moon}}}{W_{\text{Earth}}} = \frac{M_{\text{Moon}}}{M_{\text{Earth}}} \times \left(\frac{R_{\text{Earth}}}{R_{\text{Moon}}}\right)^2
\]
4. **Substitute Known Values**:
\[
W_{\text{Moon}} = W_{\text{Earth}} \times \left(\frac{1/80}{1} \times \left(\frac{4}{1}\right)^2\right)
\]
Simplify the equation:
\[
W_{\text{Moon}} = W_{\text{Earth}} \times \left(\frac{1}{80} \times
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