Question
The planet Mars has two moons, Deimos and Phobos. Phobos is about 9,000 km from Mars and is about six times more massive than Deimos. Deimos is about 23,000 km from Mars. Upon which moon does Mars exert a stronger gravitational force?
A.
Deimos and Phobos equally
B.
Phobos
C.
not enough information
D.
Deimos
A.
Deimos and Phobos equally
B.
Phobos
C.
not enough information
D.
Deimos
Answers
GPT-4o mini
The gravitational force that Mars exerts on each moon can be determined by the formula for gravitational force, which is given by:
\[ F = \frac{G \cdot m_1 \cdot m_2}{r^2} \]
where:
- \( F \) is the gravitational force,
- \( G \) is the gravitational constant,
- \( m_1 \) is the mass of Mars,
- \( m_2 \) is the mass of the moon (Deimos or Phobos in this case),
- \( r \) is the distance between the center of Mars and the center of the moon.
Given:
- Phobos is about 9,000 km from Mars and is about six times more massive than Deimos.
- Deimos is about 23,000 km from Mars.
1. **Calculating the gravitational force on Phobos:**
- The distance \( r_{Phobos} = 9,000 \, \text{km} \)
- The mass of Phobos \( m_{Phobos} = 6 \cdot m_{Deimos} \)
\[ F_{Phobos} \propto \frac{m_{Mars} \cdot (6 \cdot m_{Deimos})}{(9,000)^2} \]
2. **Calculating the gravitational force on Deimos:**
- The distance \( r_{Deimos} = 23,000 \, \text{km} \)
\[ F_{Deimos} \propto \frac{m_{Mars} \cdot m_{Deimos}}{(23,000)^2} \]
Now, we need to compare \( F_{Phobos} \) and \( F_{Deimos} \).
**Relative comparison:**
- Since the mass of Phobos is about six times that of Deimos, and it is much closer to Mars, we can assess the two forces in terms of their ratios.
Given that gravitational force is inversely proportional to the square of the distance, we can see:
- \( (9,000)^2 \) will yield a smaller denominator, resulting in a larger value for the force exerted by Mars on Phobos, relative to the denominator \( (23,000)^2 \) for Deimos.
Even though Phobos is less massive, the closer distance of Phobos greatly increases the strength of the gravitational force compared to the more distant and less massive Deimos.
Therefore, the answer is **B. Phobos**. Mars exerts a stronger gravitational force on Phobos.
\[ F = \frac{G \cdot m_1 \cdot m_2}{r^2} \]
where:
- \( F \) is the gravitational force,
- \( G \) is the gravitational constant,
- \( m_1 \) is the mass of Mars,
- \( m_2 \) is the mass of the moon (Deimos or Phobos in this case),
- \( r \) is the distance between the center of Mars and the center of the moon.
Given:
- Phobos is about 9,000 km from Mars and is about six times more massive than Deimos.
- Deimos is about 23,000 km from Mars.
1. **Calculating the gravitational force on Phobos:**
- The distance \( r_{Phobos} = 9,000 \, \text{km} \)
- The mass of Phobos \( m_{Phobos} = 6 \cdot m_{Deimos} \)
\[ F_{Phobos} \propto \frac{m_{Mars} \cdot (6 \cdot m_{Deimos})}{(9,000)^2} \]
2. **Calculating the gravitational force on Deimos:**
- The distance \( r_{Deimos} = 23,000 \, \text{km} \)
\[ F_{Deimos} \propto \frac{m_{Mars} \cdot m_{Deimos}}{(23,000)^2} \]
Now, we need to compare \( F_{Phobos} \) and \( F_{Deimos} \).
**Relative comparison:**
- Since the mass of Phobos is about six times that of Deimos, and it is much closer to Mars, we can assess the two forces in terms of their ratios.
Given that gravitational force is inversely proportional to the square of the distance, we can see:
- \( (9,000)^2 \) will yield a smaller denominator, resulting in a larger value for the force exerted by Mars on Phobos, relative to the denominator \( (23,000)^2 \) for Deimos.
Even though Phobos is less massive, the closer distance of Phobos greatly increases the strength of the gravitational force compared to the more distant and less massive Deimos.
Therefore, the answer is **B. Phobos**. Mars exerts a stronger gravitational force on Phobos.