determine the mass of the earth from the known period (365.25 days) and the distance of the moon (3.84X10^3 km)
The gravitational force of attraction of the Sun on the Earth, or the Earth on the Sun, is given by Newtons Universal Law of Gravitation
F = GMm/r^2
where F is the attractive force, G is the Universal Gravitational Constant, 1.069304x10^-9, M is the mass of the Sun, m is the mass of the earth, and r is the distance between them. This force of attraction of the Sun on the earth is what causes the centripetal acceleration which constantly accelerates the Earth toward the Sun, keeping it in its orbit at a mean distance of 92,960,242 miles. This centripetal force is expressed by F = mV^2/r where m is the mass of the earth, V is the orbital velocity of the Earth, and r the radial distance of the Earth from the Sun.
Since these two forces are equal to one another, we may write
F = GMm/r^2 = mV^2/r
from which we derive the mass of the Sun as
M(S) = rV^2/G.
Taking the mean distance between the Earth and the Sun as 92,960,242 miles, and the Earth's mean orbital velocity of 07,741 ft/sec., we can write for the mass of the Sun
M(S) = [(92,960,242(5280))97,741^2]/1.069304x10^-9 = 4.385x10^30 lb. mass.
Applying the same logic to the known information about the Earth and Moon:
The Moons mean orbital velocity is ~3340 ft/sec. while its mean distance may be taken as ~238,868 miles. Therefore, for the mass of the Earth we get
m(E) = [(238,868(5280))3340^2]/1.069304x10^-9 = 1.315x10^30 lbs. mass.
Now apply the same logic to your specific problem.
To determine the mass of the earth using the known period (365.25 days) and the distance of the moon (3.84 × 10^3 km), we can apply Newton's law of universal gravitation. This law states that the gravitational force between two objects (in this case, the Earth and the Moon) is proportional to the product of their masses and inversely proportional to the square of the distance between them.
The equation for gravitational force is:
F = (G * m1 * m2) / r^2
Where:
F is the gravitational force
G is the gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2)
m1 and m2 are the masses of the two objects
r is the distance between the centers of the two objects
We know the following values:
- Period of the Earth around the Sun (365.25 days or 3.15576 × 10^7 seconds)
- Distance between the Earth and the Moon (3.84 × 10^3 km or 3.84 × 10^6 meters)
To find the mass of the Earth (m1), we need to rearrange the formula:
m1 = ((F * r^2) / G) / m2
Now, the force acting between the Earth and the Moon (F) can be calculated using Newton's second law of motion:
F = m2 * (4π^2 * r) / T^2
Where:
T is the period of the Moon around the Earth (365.25 days or 3.15576 × 10^7 seconds)
r is the distance between the centers of the Earth and the Moon (3.84 × 10^6 meters)
m2 is the mass of the Moon
By substituting this expression for F back into the previous equation, we can then calculate the mass of the Earth (m1).
m1 = (((m2 * (4π^2 * r) / T^2) * r^2) / G) / m2
Simplifying the equation:
m1 = (4π^2 * r^3) / (G * T^2)
Now, let's substitute the known values into the equation:
m1 = (4π^2 * (3.84 × 10^6)^3) / (6.67430 × 10^-11 * (3.15576 × 10^7)^2)
Calculating this expression will give us the mass of the Earth.