The Earth has a mass of 5.97 x10^24 kg and the Moon has a mass of 7.36 x10^22kg. The average distance between the center of the Earth and the center of the moon is 3.84 x10^8m.
(a) How far away would the Moon need to be for the magnitude of the gravitational force acting on it due to the Earth be the same as the magnitude of the gravitational force of a 72kg student sitting at their desk on the surface of the earth?
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To determine the distance at which the gravitational force exerted by the Earth on the Moon is equal to the gravitational force exerted by a 72 kg student on the surface of the Earth, we can use the formula for the gravitational force:
F = G * (m1 * m2) / r^2
Where:
F is the gravitational force,
G is the gravitational constant (approximately 6.674 x 10^-11 N(m/kg)^2),
m1 and m2 are the masses of the two objects, and
r is the distance between their centers.
First, let's calculate the gravitational force between the Earth and the student on Earth's surface. We'll assume the mass of the Earth is concentrated at its center.
Mass of the Earth (m1) = 5.97 x 10^24 kg
Mass of the student (m2) = 72 kg
Distance between the Earth's center and the student (r) = radius of the Earth = 6.37 x 10^6 m
Plugging these values into the formula:
F = (6.674 x 10^-11 N(m/kg)^2) * ((5.97 x 10^24 kg) * (72 kg)) / (6.37 x 10^6 m)^2
F = 9.803 N (approximately)
The gravitational force between the Earth and the student is approximately 9.803 N.
Next, let's calculate the distance at which the gravitational force between the Earth and the Moon is equal to 9.803 N.
Mass of the Moon (m1) = 7.36 x 10^22 kg
Distance between the Earth's center and the Moon's center (r) = 3.84 x 10^8 m (given)
Now we rearrange the gravitational force formula to solve for distance (r):
r = sqrt((G * m1 * m2) / F)
Plugging in the values:
r = sqrt((6.674 x 10^-11 N(m/kg)^2) * ((5.97 x 10^24 kg) * (7.36 x 10^22 kg)) / (9.803 N))
r ≈ 3.79 x 10^5 m
Therefore, the Moon needs to be approximately 3.79 x 10^5 meters (or 379 km) away from the Earth for the magnitude of the gravitational force acting on it due to the Earth to be the same as the magnitude of the gravitational force of a 72 kg student sitting at their desk on the surface of the Earth.
To determine the distance at which the magnitude of the gravitational force acting on the Moon due to the Earth is equal to the magnitude of the gravitational force on a 72kg student on the surface of the Earth, you can use Newton's law of universal gravitation:
F_gravity = G * (m1 * m2) / r^2
Where:
F_gravity is the gravitational force
G is the universal gravitational constant (approximately 6.674 x 10^-11 Nm^2/kg^2)
m1 and m2 are the masses of the two objects
r is the distance between the centers of the two objects
Let's denote the distance we want to find as "d". The mass of the Moon (m1) is 7.36 x 10^22 kg, and the mass of the Earth (m2) is 5.97 x 10^24 kg. The mass of the student (m2') is 72 kg.
The gravitational force acting on the Moon due to the Earth is equal to the gravitational force acting on the student on the surface of the Earth. So, we can write:
G * (m1 * m2) / (3.84 x 10^8)^2 = G * (m2' * m2) / r^2
Simplifying the equation by plugging in the known values and solving for "d":
((7.36 x 10^22) * (5.97 x 10^24)) / (3.84 x 10^8)^2 = (72 * (5.97 x 10^24)) / d^2
Now, we can solve for "d" by cross-multiplying and taking the square root:
d^2 = ((72 * (5.97 x 10^24)) * (3.84 x 10^8)^2) / (7.36 x 10^22)
d = sqrt(((72 * (5.97 x 10^24)) * (3.84 x 10^8)^2) / (7.36 x 10^22))
Evaluating this expression will give you the distance "d" that the Moon needs to be for the magnitude of the gravitational force acting on it due to the Earth to be equal to the magnitude of the gravitational force on the student at their desk on the surface of the Earth.