Use Newton's Law of Gravitation to calculate the work, W, required to propel an 800 kg satellite out of Earth's gravitational field. Assume that Earth's mass is 5.98 x 10^24 kg and is concentrated at its center. The radius of Earth is 6.37 x 10^6 m, and the gravitational constant, G, is 6.67 x 10^-11 Nm^2/kg^2.
Physics · High School · Thu Feb 04 2021
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To compute the work W required to propel a satellite out of Earth's gravitational field, we essentially need to find the energy required to move the satellite from Earth's surface to a point at an infinite distance away where the gravitational force is negligible.
The force of gravity F between two masses m1 and m2 separated by a distance r is given by Newton's Law of Gravitation as:
\[ F = \frac{G \cdot m1 \cdot m2}{r^2} \]
where G is the gravitational constant.
The work done W moving a satellite from the Earth's surface to a point infinitely far away is equal to the gravitational potential energy U at the surface, which can be calculated using the formula:
\[ U = - \frac{G \cdot M \cdot m}{r} \]
where M is the mass of Earth, m is the mass of the satellite, and r is the radius of Earth. The negative sign indicates that the force of gravity is attractive.
Now, to escape Earth's gravity, the work done would be opposite to the gravitational potential energy at Earth's surface (hence why we don't actually need the negative sign in practical calculations because we're looking for the magnitude of work).
\[ W = \frac{G \cdot M \cdot m}{r} \]
Using the values provided: \[ M = 5.98 \times 10^{24} \text{kg} \] \[ m = 800 \text{kg} \] \[ r = 6.37 \times 10^{6} \text{m} \] \[ G = 6.67 \times 10^{-11} \text{Nm}^2/\text{kg}^2 \]
\[ W = \frac{(6.67 \times 10^{-11} \text{Nm}^2/\text{kg}^2) \cdot (5.98 \times 10^{24} \text{kg}) \cdot (800 \text{kg})}{6.37 \times 10^{6} \text{m}} \]
\[ W = \frac{(6.67 \times 10^{-11}) \cdot (5.98 \times 10^{24}) \cdot 800}{6.37 \times 10^{6}} \]
\[ W \approx \frac{3.2 \times 10^{14}}{6.37 \times 10^6} \]
\[ W \approx 5.02 \times 10^{7} \text{J} \]
Therefore, the work required to propel the 800 kg satellite out of Earth's gravitational field is approximately \(5.02 \times 10^{7} \text{Joules}\).