Starting from Newton’s law of universal gravitation, show how to find the speed of the moon in its orbit from the earth-moon distance of 3.9 × 108 m and the earth’s mass. Assume the orbit is a circle.
Let V be the velocity. Assume the earth's velocity is much larger and that the moon goes around the Earth at the center. Actually, both revolve about the center of mass of the pair, at a speed that depends upon the sum of the masses.
Let M be the Earth's mass.
R = 3.9*10^8 m
G = universal constant of gravity
GM/R^2 = V^2/R
Solve for V
To find the speed of the moon in its orbit around the Earth using Newton's law of universal gravitation, we can use the concept of centripetal force.
Newton's law of universal gravitation states that the gravitational force between two objects is given by:
F = (G * m1 * m2) / r^2
Where:
F is the gravitational force,
G is the gravitational constant (6.67430 × 10^-11 N m^2 / kg^2),
m1 and m2 are the masses of the two objects,
and r is the distance between the centers of the two objects.
In this case, we are considering the moon to be orbiting the Earth, so the Earth is the central object. We need to rearrange the equation to solve for the speed of the moon.
Taking the equation for the gravitational force and equating it to the centripetal force (Fc = m * v^2 / r), we get:
(moon's mass * v^2) / r = G * (Earth's mass * moon's mass) / r^2
Since we are assuming the orbit is a circle, we know that the centripetal force is given by Fc = (moon's mass * v^2) / r.
Simplifying the equation further, we have:
v^2 = (G * Earth's mass) / r
To find the speed of the moon (v), we need to take the square root of both sides of the equation:
v = sqrt((G * Earth's mass) / r)
Now, we can substitute the given values into the equation. Assume the Earth's mass is M and the distance between the Earth and the moon is r.
Given:
Earth's mass (M) = known value
Distance (r) = 3.9 × 10^8 m
Substituting these values in, the equation becomes:
v = sqrt((G * M) / r)
Substituting the known values for G and r, we can calculate the speed of the moon in its orbit around the Earth.