7. A satellite with a mass of 350 kg is traveling by Saturn \( \left(\mathrm{m}=5.68 \times 10^{26} \mathrm{~kg}\right) \) at a distance of \( 8 \times 10^{7} \mathrm{~m} \) from its center. (A) Determine the magnitude of the force of gravity acting on the satellite due to Saturn \( F_{g}=m g \) \( =350 \)

To determine the magnitude of the gravitational force acting on the satellite due to Saturn, we can use Newton's law of universal gravitation, which is given by the formula: \[ F_g = \frac{G \cdot m_1 \cdot m_2}{r^2} \] where: - \( F_g \) is the gravitational force, - \( G \) is the gravitational constant, approximately \( 6.674 \times 10^{-11} \, \text{m}^3/\text{kg} \cdot \text{s}^2 \), - \( m_1 \) is the mass of Saturn (\( 5.68 \times 10^{26} \, \text{kg} \)), - \( m_2 \) is the mass of the satellite (\( 350 \, \text{kg} \)), - \( r \) is the distance from the center of Saturn to the satellite (\( 8 \times 10^{7} \, \text{m} \)). Now, we can substitute the known values into the formula to calculate the gravitational force \( F_g \). Let's perform the calculation step by step: 1. Substitute the values into the formula: \[ F_g = \frac{(6.674 \times 10^{-11}) \cdot (5.68 \times 10^{26}) \cdot (350)}{(8 \times 10^{7})^2} \] 2. Calculate \( (8 \times 10^{7})^2 \): \[ (8 \times 10^{7})^2 = 64 \times 10^{14} = 6.4 \times 10^{15} \] 3. Now substitute this back into the equation: \[ F_g = \frac{(6.674 \times 10^{-11}) \cdot (5.68 \times 10^{26}) \cdot (350)}{6.4 \times 10^{15}} \] 4. Calculate the numerator: \[ (6.674 \times 10^{-11}) \cdot (5.68 \times 10^{26}) \cdot (350) = 1.33 \times 10^{16} \] 5. Now divide the numerator by the denominator: \[ F_g = \frac{1.33 \times 10^{16}}{6.4 \times 10^{15}} \approx 2.08 \, \text{N} \] Thus, the magnitude of the gravitational force acting on the satellite due to Saturn is approximately \( 2.08 \, \text{N} \).