Communications engineers attempt to place an artificial satellite of mass $m_{s}=5.19 \times 10^{3} \mathrm{~kg}$ in a geosynchronous orbit in which the satellite remains over one spot on Earth's surface at an altitude of $3.56 \times 10^{8} \mathrm{~m}$ above sea level. The mass of Earth is $M_{E}=5.9 \times 10^{24} \mathrm{~kg}$. The radius of Earth is $R_{E}=6.37 \times 10^{6} \mathrm{~m}$. The total mechanical energy of the satellite-Earth system should be for a stable geosynchronous orbit, but the engineers placed this satellite in an orbit at the correct radius but with a total mechanical energy of $E=-2.46 \times 10^{10} \mathrm{~J}$. Which of the following is a correct value for the kinetic energy of the satellite along with a possible result from having the incorrect energy? A $K=8.15 \times 10^{6} \mathrm{~J}$; the satellite's orbit will decay as it spirals into Earth's surface. B $K=2.46 \times 10^{10} \mathrm{~J}$; the satellite will enter an elliptical orbit about Earth with a smaller average radius than desired. (C) $K=2.46 \times 10^{10} \mathrm{~J}$; the kinetic energy is close enough to the correct energy, so the satellite will remain in the desired orbit. (D) $K=4.87 \times 10^{10} \mathrm{~J}$; the satellite will enter an elliptical orbit about Earth with a larger average radius than desired. (E) $K=2.46 \times 10^{13} \mathrm{~J}$; the satellite will leave Earth's orbit.
A geosynchronous satellite orbits above the equator of Earth in a circular orbit, remaining in a fixed position with respect to observers on the ground. Therefore, the satellite's period of revolution must be equal to the period of rotation of the Earth. When calculations are made to launch a certain geosynchronous satellite and the satellite is launched, the satellite's orbital period is slightly larger than the rotational period of Earth. Which of the following could have caused the difference? A The mass of the satellite was larger than what was used in the calculation. B The mass of the satellite was smaller than what was used in the calculation. C The altitude of the satellite was larger than predicted by the calculation. D The altitude of the satellite was smaller than predicted by the calculation. E The diameter of the satellite was larger than what was used in the calculation.
Note: Figure not drawn to scale.
A planet of mass $M$ and radius $R$ has no atmosphere. The escape velocity at its surface is $v_{e}$. An object of mass $m$ is at rest a distance $r$ from the center of the planet, where $r>>R$. The particle falls to the surface of the planet. The total mechanical energy of the particle at the surface of the planet is closest to A 0 (B) $\frac{1}{2} m v_{e}^{2}$ (C) $-G \frac{m M}{R}$ (D) $+G \frac{m M}{R}$ (E) $\frac{1}{2} m v_{e}^{2}+G \frac{m M}{R}$