Darwin proposed that the Moon formed far closer to Earth than it is now. Let’s see just how close it could possibly be to Earth and survive. In particular, in the next 4 problems you will find the distance from Earth at which the tidal force due to Earth is sufficient to lift a rock off the lunar surface. To do this, imagine a rock of mass m on the lunar surface. Write an expression for the gravitational force applied by the Moon to the rock, in terms of the lunar mass Mm (input as M_m) and the lunar radius Rm (input as R_m).

Question

Darwin proposed that the Moon formed far closer to Earth than it is now. Let’s see just how close it could possibly be to Earth and survive. In particular, in the next 4 problems you will find the distance from Earth at which the tidal force due to Earth is sufficient to lift a rock off the lunar surface. To do this, imagine a rock of mass m on the lunar surface. Write an expression for the gravitational force applied by the Moon to the rock, in terms of the lunar mass Mm (input as M_m) and the lunar radius Rm (input as R_m).

Answer 1

Let the moon rock mass be m

F = G*m*(M_m)/(R_m)^2
is the force that hold the rock to the moon. It is the same as its weight.

Learn the universal law of gravity.

Answer 2

To find the gravitational force applied by the Moon to the rock on the lunar surface, we can use Newton's law of universal gravitation. The equation is:

F = G * (Mm * m) / (Rm^2)

Where:
F is the gravitational force applied by the Moon to the rock.
G is the gravitational constant.
Mm is the mass of the Moon.
m is the mass of the rock.
Rm is the radius of the Moon.

So, the expression for the gravitational force applied by the Moon to the rock is:

F = G * (Mm * m) / (Rm^2)

To find the distance from Earth at which the tidal force due to Earth is sufficient to lift the rock off the lunar surface, we need to consider the tidal force. The tidal force arises due to the difference in gravitational force on the near and far sides of an object caused by another massive object.

In this case, the tidal force due to Earth is given by:

F_tidal = 2 * G * (Mm * m) * (Rm / D)^3

Where:
F_tidal is the tidal force.
D is the distance between Earth and the Moon.

To find the distance at which the tidal force is sufficient to lift the rock off the lunar surface, we set the gravitational force equal to the tidal force:

F = F_tidal

G * (Mm * m) / (Rm^2) = 2 * G * (Mm * m) * (Rm / D)^3

Dividing both sides of the equation by G * (Mm * m), we get:

1 / (Rm^2) = 2 * (Rm / D)^3

Now, we can solve for D:

( Rm / D)^3 = 1 / (2 * Rm^2)

Taking the cube root of both sides:

Rm / D = (1 / (2 * Rm^2))^(1/3)

Multiplying both sides by D:

Rm = D * (1 / (2 * Rm^2))^(1/3)

D = Rm / ((1 / (2 * Rm^2))^(1/3))

Therefore, the distance from Earth at which the tidal force due to Earth is sufficient to lift a rock off the lunar surface is:

D = Rm / ((1 / (2 * Rm^2))^(1/3))

You can plug in the values of Mm and Rm to calculate the distance D.