If the period of the moon were 31 days while its orbit radius 3.8x10^8m, what would the mass of the Earth be?
31 days (24 hr/day)(3600 s/hr) = 2.68*10^6 seconds
circumference of orbit = 2 pi r = 23.9*10^8
so v, speed = 23.9*10^8/(2.68*10^6
= 8.91*10^2 m/s
Ac = v^2/r = 79.4 * 10^4 / 3.8*10^8
= 20.9 *10^-4 m/s^2
F = m a
G Me m/r^2 = m Ac note m moon cancels
6.67*10^-11 Me/14.4*10^16 =20.9*10^-4
Me = 45.2 *10^23 = 4.52*10^24 Kg
check my arithmetic but that is the right order of magnitude.
Where do you get the 14.4*10^16 from?
orbit radius squared
To calculate the mass of the Earth based on the given information, we can use Newton's law of universal gravitation, which states:
F = (G * (m1 * m2)) / r²
Where:
- F is the gravitational force between two objects.
- G is the gravitational constant (approximately 6.67430 x 10^-11 N(m/kg)²).
- m1 and m2 are the masses of the two objects.
- r is the distance between the centers of the two objects.
In this case, the Earth and the Moon are the two objects. We know the period of the moon (T), which is the time it takes to complete one orbit around the Earth, and we also know the radius of the moon's orbit (r).
The period of the moon (T) is related to the radius of the orbit (r) by the following equation:
T = 2π * √(r³ / (G * M))
Where:
- T is the period of the moon.
- π is the mathematical constant pi (approximately 3.14159).
- r is the radius of the moon's orbit.
- G is the gravitational constant.
- M is the mass of the Earth.
By rearranging the equation, we can solve for M:
M = (4π² * r³) / (G * T²)
Now we can substitute the given values and solve for the mass of the Earth:
G = 6.67430 x 10^-11 N(m/kg)²
T = 31 days = 31 * 24 * 3600 seconds (convert to seconds)
r = 3.8 x 10^8 meters
First, let's convert the period of the moon from days to seconds:
T = 31 * 24 * 3600 seconds = 2,678,400 seconds
Now we substitute the values into the equation and calculate the mass of the Earth:
M = (4π² * (3.8 x 10^8)³) / (6.67430 x 10^-11 * (2,678,400)²)
Calculating this expression will give us the mass of the Earth.