Exercises: Motion of Orbiting Satellites

A satellite moves in a circular orbit of radius $7.0\times10^{6}\ \text{m}$ measured from Earth's center. Use $M_E = 5.97\times10^{24}\ \text{kg}$ and $G = 6.67\times10^{-11}\ \text{N·m}^2/\text{kg}^2$. Calculate the satellite's orbital speed. Does the answer depend on the satellite's mass?

A satellite in a circular orbit around Earth is boosted to a new circular orbit of larger radius. For each quantity, state whether it increases, decreases, or stays the same, and justify your choice briefly: (a) orbital speed, (b) orbital period, (c) gravitational force on the satellite.

The Moon orbits Earth in a nearly circular path of radius $3.84\times10^{8}\ \text{m}$ with a period of $27.3$ days. Convert the period to seconds and calculate the Moon's average orbital speed. Give your answer in m/s.

A communications satellite is placed in a geostationary orbit, so it stays above the same point on the equator. Its period is $8.64\times10^{4}\ \text{s}$. Use $M_E = 5.97\times10^{24}\ \text{kg}$, $R_E = 6.37\times10^{6}\ \text{m}$, and $G = 6.67\times10^{-11}\ \text{N·m}^2/\text{kg}^2$. (a) Derive an expression for the orbital radius in terms of $G$, $M_E$, and the period. (b) Calculate the orbital radius and the altitude above Earth's surface. (c) Explain why a geostationary orbit must lie above the equator.

Satellite A and satellite B have the same mass and both orbit Earth in circular orbits. Satellite A has orbital radius $r$ and satellite B has orbital radius $2r$. Find the ratio of each quantity, A to B: (a) orbital speed, (b) orbital period, (c) kinetic energy. Show how each ratio follows from the orbit equations.

A probe orbits Mars in a circular orbit $4.00\times10^{5}\ \text{m}$ above the surface. Mars has mass $6.42\times10^{23}\ \text{kg}$ and radius $3.39\times10^{6}\ \text{m}$. Use $G = 6.67\times10^{-11}\ \text{N·m}^2/\text{kg}^2$. (a) Calculate the orbital radius measured from Mars's center. (b) Calculate the probe's orbital speed. (c) Calculate the orbital period in minutes.

A satellite moves in an elliptical orbit around Earth. Its closest point (perigee) is at distance $7.0\times10^{6}\ \text{m}$ and its farthest point (apogee) is at distance $2.1\times10^{7}\ \text{m}$, both measured from Earth's center. Use $M_E = 5.97\times10^{24}\ \text{kg}$ and $G = 6.67\times10^{-11}\ \text{N·m}^2/\text{kg}^2$. (a) Use conservation of angular momentum to relate the speeds at perigee and apogee. (b) Combine this with conservation of energy to find both speeds. (c) Compare the perigee speed with the speed of a circular orbit at the same radius, and explain what the comparison tells you about the orbit's shape.

A satellite of mass $1{,}200\ \text{kg}$ is moved from a circular orbit of radius $7.0\times10^{6}\ \text{m}$ to a circular orbit of radius $1.4\times10^{7}\ \text{m}$ around Earth. Use $M_E = 5.97\times10^{24}\ \text{kg}$, $G = 6.67\times10^{-11}\ \text{N·m}^2/\text{kg}^2$, and $U = -GM_Em/r$. (a) Calculate the change in kinetic energy, in gravitational potential energy, and in total mechanical energy. (b) Determine the minimum net work the engines must do. (c) Explain why the satellite's speed decreases even though the engines did positive work on it.