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How much work is required to lift a $1000-\mathrm{kg}$ satellite from the surface of the earth to an altitude of $2 \cdot 10^{6} \mathrm{m} ?$ The gravitational for...

Question

How much work is required to lift a $1000-\mathrm{kg}$ satellite from the surface of the earth to an altitude of $2 \cdot 10^{6} \mathrm{m} ?$ The gravitational force is $F=G M m / r^{2},$ where $M$ is the mass of the earth, $m$ is the mass of the satellite, and $r$ is the distance between them. The radius of the earth is $6.4 \cdot 10^{6} \mathrm{m},$ its mass is $6 \cdot 10^{24} \mathrm{kg},$ and in these units the gravitational constant, $G,$ is $6.67 \cdot 10^{-11}$

How much work is required to lift a $1000-\mathrm{kg}$ satellite from the surface of the earth to an altitude of $2 \cdot 10^{6} \mathrm{m} ?$ The gravitational force is $F=G M m / r^{2},$ where $M$ is the mass of the earth, $m$ is the mass of the satellite, and $r$ is the distance between them. The radius of the earth is $6.4 \cdot 10^{6} \mathrm{m},$ its mass is $6 \cdot 10^{24} \mathrm{kg},$ and in these units the gravitational constant, $G,$ is $6.67 \cdot 10^{-11}$



Answers

(a) Newton's Law of Gravitation states that two bodies with masses $ m_1 $ and $ m_2 $ attract each other with a force
$$ F = G \frac{m_1 m_2}{r^2} $$
where $ r $ is the distance between the bodies and $ G $ is the gravitational constant. If one of the bodies is fixed, find the work needed to move the other from $ r = a $ to $ r = b $.
(b) Compute the work required to launch a 1000-kg satellite vertically to a height of 1000 km. You may assume that the earth's mass is $ 5.98 \times 10^{24} kg $ and is concentrated at its center. Take the radius of the earth to be $ 6.37 \times 10^6 m $ and $ G = 6.67 \times 10^{-11} N \cdot m^2/kg^2 $.


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