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8. Centripetal Forces & Gravitation
Newton's Law of Gravity
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Assume that you stay on the Earth's surface. What is the ratio of the Sun's gravitational force on you to the Earth's gravitational force on you?
A
Approximately 1/2
B
Approximately 1/100
C
Approximately 1/10
D
Approximately 1/1600
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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.
Calculate the gravitational force exerted by the Earth: Use the formula \( F_{\text{Earth}} = \frac{G \cdot m_{\text{Earth}} \cdot m_{\text{you}}}{r_{\text{Earth}}^2} \), where \( m_{\text{Earth}} \) is the mass of the Earth, \( m_{\text{you}} \) is your mass, and \( r_{\text{Earth}} \) is the radius of the Earth.
Calculate the gravitational force exerted by the Sun: Use the formula \( F_{\text{Sun}} = \frac{G \cdot m_{\text{Sun}} \cdot m_{\text{you}}}{r_{\text{Sun}}^2} \), where \( m_{\text{Sun}} \) is the mass of the Sun, \( m_{\text{you}} \) is your mass, and \( r_{\text{Sun}} \) is the average distance from the Earth to the Sun.
Determine the ratio of the Sun's gravitational force to the Earth's gravitational force: The ratio is given by \( \frac{F_{\text{Sun}}}{F_{\text{Earth}}} = \frac{m_{\text{Sun}} / r_{\text{Sun}}^2}{m_{\text{Earth}} / r_{\text{Earth}}^2} \). Notice that \( G \) and \( m_{\text{you}} \) cancel out in the ratio.
Simplify the expression: Substitute the known values for \( m_{\text{Sun}} \), \( m_{\text{Earth}} \), \( r_{\text{Sun}} \), and \( r_{\text{Earth}} \) to find the approximate ratio, which is approximately 1/1600.
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