Question

Find the following center-of-mass information about objects in the Solar System. You can look up the necessary data on the Internet or in the tables in Chapter 12 of this book. Assume spherically symmetrical mass distributions for all objects under consideration. a) Determine the distance from the center of mass of the Earth-Moon system to the geometric center of Earth. b) Determine the distance from the center of mass of the Sun-Jupiter system to the geometric center of the Sun.

Short answer

Answer: (a) The distance from the center of mass of the Earth-Moon system to the geometric center of Earth is approximately 4670 km. (b) The distance from the center of mass of the Sun-Jupiter system to the geometric center of the Sun is approximately 742,000 km.

Step-by-step solution

a) Distance from the center of mass of the Earth-Moon system to the geometric center of Earth

: Step 1: Find the masses of Earth and Moon Look up the masses of Earth (m_Earth) and Moon (m_Moon). m_Earth = 5.972 x 10^24 kg m_Moon = 7.342 x 10^22 kg Step 2: Find the distance between Earth and Moon Look up the average distance between Earth and Moon (r_EarthMoon). r_EarthMoon = 384,400 km Step 3: Calculate the center of mass for the Earth-Moon system Apply the center of mass formula: COM = (m1 * r1 + m2 * r2) / (m1 + m2) Since we want to find the distance from the center of mass to the geometric center of Earth, we choose r1 = 0 (Earth as our reference point). Therefore, r2 = r_EarthMoon. COM = (m_Earth * 0 + m_Moon * r_EarthMoon) / (m_Earth + m_Moon) Step 4: Calculate the result Plug in the values and calculate the distance: COM = (7.342 x 10^22 kg * 384,400 km) / (5.972 x 10^24 kg + 7.342 x 10^22 kg) COM ≈ 4670 km The distance from the center of mass of the Earth-Moon system to the geometric center of Earth is approximately 4670 km.

b) Distance from the center of mass of the Sun-Jupiter system to the geometric center of the Sun

: Step 1: Find the masses of the Sun and Jupiter Look up the masses of Sun (m_Sun) and Jupiter (m_Jupiter). m_Sun = 1.989 x 10^30 kg m_Jupiter = 1.898 x 10^27 kg Step 2: Find the distance between Sun and Jupiter Look up the average distance between Sun and Jupiter (r_SunJupiter). r_SunJupiter = 778.5 x 10^6 km Step 3: Calculate the center of mass for the Sun-Jupiter system Apply the center of mass formula: COM = (m1 * r1 + m2 * r2) / (m1 + m2) Since we want to find the distance from the center of mass to the geometric center of the Sun, we choose r1 = 0 (Sun as our reference point). Therefore, r2 = r_SunJupiter. COM = (m_Sun * 0 + m_Jupiter * r_SunJupiter) / (m_Sun + m_Jupiter) Step 4: Calculate the result Plug in the values and calculate the distance: COM = (1.898 x 10^27 kg * 778.5 x 10^6 km) / (1.989 x 10^30 kg + 1.898 x 10^27 kg) COM ≈ 742,000 km The distance from the center of mass of the Sun-Jupiter system to the geometric center of the Sun is approximately 742,000 km.