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Show that Titan's surface gravity is about one-seventh that of Earth's.

Show that Titan's surface gravity is about one-seventh that of Earth's.

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**Demonstrating Titan's Surface Gravity in Relation to Earth's** To show that Titan’s surface gravity is about one-seventh that of Earth’s, we need to compare the gravitational forces on the surface of both celestial bodies. ### Understanding Gravity Gravity is the force that attracts a body toward the center of the Earth, or any other physical body having mass. The surface gravity of a celestial body is dependent on its mass and radius. The formula to calculate surface gravity (\( g \)) is given by: \[ g = \frac{GM}{r^2} \] Where: - \( G \) is the gravitational constant, - \( M \) is the mass of the celestial body, - \( r \) is the radius of the celestial body. ### Earth's Surface Gravity Earth's surface gravity \( g_E \) is approximately 9.81 \( m/s^2 \). ### Titan’s Surface Gravity Titan is the largest moon of Saturn, and its physical characteristics contribute to its lower surface gravity. Titan has a mass \( M_T \) and a radius \( r_T \) which are different from those of Earth. 1. **Mass and Radius of Titan**: - Mass of Titan (\( M_T \)): \( 1.3452 \times 10^{23} \) kg - Radius of Titan (\( r_T \)): 2,575 km 2. **Mass and Radius of Earth**: - Mass of Earth (\( M_E \)): \( 5.97237 \times 10^{24} \) kg - Radius of Earth (\( r_E \)): 6,371 km ### Calculation 1. Using the formula for surface gravity: \[ g_T = \frac{GM_T}{r_T^2} \] 2. Substitute the respective values for Titan: - \( G = 6.67430 \times 10^{-11} \) m\(^3\)kg\(^{-1}\)s\(^{-2}\) - \( M_T = 1.3452 \times 10^{23} \) kg - \( r_T = 2,575,000 \) m \[ g_T = \frac{(6.67430 \times 10^{-11})(1.3452 \times 10^{23})}{(2,575,000)^2} \] Performing the calculation
Transcribed Image Text:**Demonstrating Titan's Surface Gravity in Relation to Earth's** To show that Titan’s surface gravity is about one-seventh that of Earth’s, we need to compare the gravitational forces on the surface of both celestial bodies. ### Understanding Gravity Gravity is the force that attracts a body toward the center of the Earth, or any other physical body having mass. The surface gravity of a celestial body is dependent on its mass and radius. The formula to calculate surface gravity (\( g \)) is given by: \[ g = \frac{GM}{r^2} \] Where: - \( G \) is the gravitational constant, - \( M \) is the mass of the celestial body, - \( r \) is the radius of the celestial body. ### Earth's Surface Gravity Earth's surface gravity \( g_E \) is approximately 9.81 \( m/s^2 \). ### Titan’s Surface Gravity Titan is the largest moon of Saturn, and its physical characteristics contribute to its lower surface gravity. Titan has a mass \( M_T \) and a radius \( r_T \) which are different from those of Earth. 1. **Mass and Radius of Titan**: - Mass of Titan (\( M_T \)): \( 1.3452 \times 10^{23} \) kg - Radius of Titan (\( r_T \)): 2,575 km 2. **Mass and Radius of Earth**: - Mass of Earth (\( M_E \)): \( 5.97237 \times 10^{24} \) kg - Radius of Earth (\( r_E \)): 6,371 km ### Calculation 1. Using the formula for surface gravity: \[ g_T = \frac{GM_T}{r_T^2} \] 2. Substitute the respective values for Titan: - \( G = 6.67430 \times 10^{-11} \) m\(^3\)kg\(^{-1}\)s\(^{-2}\) - \( M_T = 1.3452 \times 10^{23} \) kg - \( r_T = 2,575,000 \) m \[ g_T = \frac{(6.67430 \times 10^{-11})(1.3452 \times 10^{23})}{(2,575,000)^2} \] Performing the calculation
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