Multiple Choice
How many kilometers above the Earth's surface must you travel for your weight to decrease to 49% of its value at the surface, according to Newton's Law of Gravity?
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8. Centripetal Forces & Gravitation
Newton's Law of Gravity
Multiple Choice
According to Newton's Law of Gravity, if the gravitational force exerted by the Sun on the Earth is 100%, what percentage of this force is exerted by the Moon on the Earth?
A
0.017%
B
17%
C
0.17%
D
1.7%
Verified step by step guidance1
Start by recalling Newton's Law of Universal Gravitation, which states that the gravitational force \( F \) between two masses \( m_1 \) and \( m_2 \) is given by \( F = \frac{G m_1 m_2}{r^2} \), where \( G \) is the gravitational constant and \( r \) is the distance between the centers of the two masses.
Identify the two scenarios: the gravitational force exerted by the Sun on the Earth and the gravitational force exerted by the Moon on the Earth. For both scenarios, the Earth is one of the masses, so we need to compare the forces exerted by the Sun and the Moon.
Recognize that the gravitational force is directly proportional to the mass of the celestial body exerting the force and inversely proportional to the square of the distance from the Earth. Therefore, the force exerted by the Sun \( F_{\text{Sun}} \) and the force exerted by the Moon \( F_{\text{Moon}} \) can be compared using the formula: \( \frac{F_{\text{Moon}}}{F_{\text{Sun}}} = \frac{m_{\text{Moon}} / r_{\text{Moon}}^2}{m_{\text{Sun}} / r_{\text{Sun}}^2} \).
Substitute the known values: the mass of the Moon \( m_{\text{Moon}} \), the mass of the Sun \( m_{\text{Sun}} \), the average distance from the Earth to the Moon \( r_{\text{Moon}} \), and the average distance from the Earth to the Sun \( r_{\text{Sun}} \).
Calculate the ratio \( \frac{F_{\text{Moon}}}{F_{\text{Sun}}} \) using the substituted values, and then convert this ratio into a percentage to find what percentage of the Sun's gravitational force is exerted by the Moon on the Earth.
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