How far from the center of the Earth (in km) is the center of mass of the Earth + satellite system if the satellite has a mass of 6560 kg and is in orbit 46200 km above the surface of the Earth. The mass of the Earth is 5.98 * 10^24 kg and the radius of the Earth is 6370 km.
Place the center of earth at x=0
totalmass*cm=Me*0+Ms*(46200+6370)
cm= Ms/(Ms+Me) * 52570 km
To find the distance of the center of mass of the Earth + satellite system from the center of the Earth, you can use the concept of the center of mass. The center of mass is the point where the entire mass of a system can be considered to be concentrated.
To begin, let's find the distance of the satellite from the center of the Earth. We know that the satellite is in orbit 46200 km above the surface of the Earth. The radius of the Earth is 6370 km. Therefore, the distance of the satellite from the center of the Earth would be:
Distance of satellite = Radius of Earth + Height of satellite
= 6370 km + 46200 km
= 52570 km
Next, we need to calculate the mass of the Earth + satellite system. The mass of the Earth is given as 5.98 * 10^24 kg, and the mass of the satellite is 6560 kg. Therefore, the total mass of the system would be:
Mass of Earth + satellite system = Mass of Earth + Mass of satellite
= 5.98 * 10^24 kg + 6560 kg
= 5.98 * 10^24 kg (since the mass of the satellite is very small compared to the mass of the Earth)
Now, we can calculate the distance of the center of mass from the center of the Earth using the formula:
Distance of center of mass = (Distance of satellite * Mass of satellite) / (Mass of Earth + Mass of satellite)
Substituting the values:
Distance of center of mass = (52570 km * 6560 kg) / (5.98 * 10^24 kg)
≈ 5.756855618 km
Therefore, the center of mass of the Earth + satellite system is approximately 5.756855618 km from the center of the Earth.