The mass of the Moon is 7.35 x 10^22 kg. At some point between Earth and the Moon, the force of Earth's gravitational attraction on an object is cancelled by the Moon's force of gravitational attraction. If the distance between Earth and the Moon (centre to centre) is 3.84 x 10^5 km, calculate where this will occur, relative to Earth.
which is the answer is correct
7.35 x 10^22
To calculate the distance where the gravitational attractions from Earth and the Moon cancel out, we can use the concept of gravitational forces.
The gravitational force between two objects is given by the equation:
F = (G * m1 * m2) / r^2
Where:
F = Gravitational force
G = Gravitational constant (6.67430 x 10^-11 N*m^2/kg^2)
m1, m2 = Masses of the two objects
r = Distance between the two objects
In this case, we want to find the distance where the forces cancel out, so the gravitational force from Earth equals the gravitational force from the Moon.
Let's assume the object has a mass of m (which cancels out in the final calculation). Also, the gravitational forces act towards the center of the respective objects, resulting in a net force of zero. So, we can set up the equation:
(G * m * M_Earth) / (d^2) = (G * m * M_Moon) / ((R - d)^2)
Where:
M_Earth = Mass of Earth
M_Moon = Mass of Moon
R = Distance between the centers of Earth and Moon (3.84 x 10^5 km)
d = Distance from Earth to the point where forces cancel out
Now, we can solve this equation to find the value of d. Let's substitute the known values:
(6.67430 x 10^-11 N*m^2/kg^2 * m * (Mass of Earth)) / (d^2) = (6.67430 x 10^-11 N*m^2/kg^2 * m * (Mass of Moon)) / ((R - d)^2)
Canceling out the mass and gravitational constant:
(Mass of Earth) / (d^2) = (Mass of Moon) / ((R - d)^2)
Now, we can rearrange the equation to solve for d:
(Mass of Earth) * ((R - d)^2) = (Mass of Moon) * (d^2)
Expanding the equation:
(R^2 - 2Rd + d^2) * (Mass of Earth) = (d^2) * (Mass of Moon)
Distributing the mass terms:
(R^2 * Mass of Earth) - (2R * d * Mass of Earth) + (d^2 * Mass of Earth) = d^2 * Mass of Moon
Rearranging the equation:
(R^2 * Mass of Earth) - (d^2 * Mass of Earth) = (2R * d * Mass of Earth) - (d^2 * Mass of Moon)
Let's substitute the known values:
(3.84 x 10^5 km)^2 * (Mass of Earth) - (d^2 * Mass of Earth) = (2 * (3.84 x 10^5 km) * d * Mass of Earth) - (d^2 * (Mass of Moon))
Now, we can solve this quadratic equation to find the value of d.
well you need to know that earth mass is about 5.98*10^24 kg
then (5.98^10^24) /x^2 = (7.35*10^22)/ (3.84*10^5 - x)^2