What is the ratio of the Sun's gravitational force on the Moon to the Earth's gravitational force on the Moon?

• A. About 1:2
• B. About 1:3
• C. About 2:1
• D. About 1:1

Answer: (C)

Verified step by step guidance

1. Identify the formula for gravitational force: The gravitational force between two masses is given by Newton's law of universal gravitation, which is \( F = \frac{G \cdot m_1 \cdot m_2}{r^2} \), where \( G \) is the gravitational constant, \( m_1 \) and \( m_2 \) are the masses, and \( r \) is the distance between the centers of the two masses.
2. Calculate the gravitational force of the Sun on the Moon: Use the formula \( F_{\text{Sun-Moon}} = \frac{G \cdot M_{\text{Sun}} \cdot M_{\text{Moon}}}{d_{\text{Sun-Moon}}^2} \), where \( M_{\text{Sun}} \) is the mass of the Sun, \( M_{\text{Moon}} \) is the mass of the Moon, and \( d_{\text{Sun-Moon}} \) is the average distance from the Sun to the Moon.
3. Calculate the gravitational force of the Earth on the Moon: Use the formula \( F_{\text{Earth-Moon}} = \frac{G \cdot M_{\text{Earth}} \cdot M_{\text{Moon}}}{d_{\text{Earth-Moon}}^2} \), where \( M_{\text{Earth}} \) is the mass of the Earth, \( M_{\text{Moon}} \) is the mass of the Moon, and \( d_{\text{Earth-Moon}} \) is the average distance from the Earth to the Moon.
4. Determine the ratio of the Sun's gravitational force on the Moon to the Earth's gravitational force on the Moon: This is given by \( \frac{F_{\text{Sun-Moon}}}{F_{\text{Earth-Moon}}} = \frac{\frac{G \cdot M_{\text{Sun}} \cdot M_{\text{Moon}}}{d_{\text{Sun-Moon}}^2}}{\frac{G \cdot M_{\text{Earth}} \cdot M_{\text{Moon}}}{d_{\text{Earth-Moon}}^2}} \).
5. Simplify the expression: Cancel out the common terms \( G \) and \( M_{\text{Moon}} \) from the numerator and the denominator, resulting in \( \frac{M_{\text{Sun}}}{M_{\text{Earth}}} \cdot \left(\frac{d_{\text{Earth-Moon}}}{d_{\text{Sun-Moon}}}\right)^2 \). Calculate this expression to find the ratio, which is approximately 2:1.