Find the orbital radius that corresponds to a "year" of 150 days.

Suppose a planet is discovered that has the same total mass as the Earth, but half its radius. Calculate the acceleration due to gravity on this planet.

Suppose the mass of the Sun is suddenly doubled, but the Earth's orbital radius remains the same. Would the length of an Earth year increase, decrease, or stay the same?

Suppose the Sun retains its present mass, but the mass of the Earth is doubled instead. Would the length of the year increase, decrease, or stay the same?

Kepler's third law states that the square of the time T required for a planet to complete one orbit around the Sun is directly proportional to the cube of the average distance d of the planet to the Sun. For the Earth assume that $d=9.3 \times 10^{7} \mathrm{mi}$ and T = 365 days. a. Find the period of Marth that the distance between Maxs and the Sun is 1.5 times the distance from the Earth to he Sun. Round to the nearest day. b. Find the average distance of Venus tothe Sun, given that Venus revolves around the Sun in 223 days. Round to the nearest million miles.

Suppose a planet is discovered that has the same total mass as the Earth, but half its radius. Is the acceleration due to gravity on this planet more than, less than, or the same as the acceleration due to gravity on the Earth? Explain.

Show that the speed of a satellite in a circular orbit a a height h above the surface of Earth is given by $v=\sqrt{\frac{G M_{\mathrm{E}}}{R_{\mathrm{E}}+h}}$

The astronomical unit $\mathrm{AU}$ is defined as the mean distance from the Sun to the Earth $\left(1 \mathrm{AU}=1.50 \times 10^{11} \mathrm{~m}\right)$. Apply Kepler's third law to the solar system, and show that it can be written as

$$T=C r^{3 / 2}$$

In this expression, the period $T$ is measured in years, the distance $r$ is measured in astronomical units, and the constant $C$ has a magnitude that you must determine.

Calculate the orbital radius of a satellite in geosynchronous orbit.

Show that the speed of a satellite in a circular orbit a height $h$ above the surface of the Earth is

$$v=\sqrt{\frac{G M_{\mathrm{E}}}{R_{\mathrm{E}}+h}}$$

It is often said that astronauts in orbit experience weightlessness because they are beyond the pull of Earth's gravity. Is this statement correct? Explain.

Suppose a planet is discovered that has the same amount of mass in a given volume as Earth, but has half Earth's radius. (a) Is the acceleration due to gravity on this planet more than, less than, or the same as the acceleration due to gravity on Earth? Explain. (b) Calculate the acceleration due to gravity on this planet. (Hint: Use $V=\frac{4}{3}(\pi) r^{3}$ 3 to calculate the volume of a sphere.)

Suppose a planet is discovered that has the same total mass as Earth, but half its radius. (a) Is the acceleration due to gravity on this planet more than, less than, or the same as the acceleration due to gravity on Earth? Explain. (b) Calculate the acceleration due to gravity on this planet.

A satellite is placed in Earth orbit at an altitude of $23,300 \mathrm{mi}$, which is 1100 miles higher than the altitude of a geosynchronous satellite. Is the period of this satellite greater than or less than 24 hours?