suppose the moon were held in its orbit not by gravitational interaction with the Earth but by a long, mass-less cable attached to the center of the earth. What would be the tension in the cable?

What is centripetal force? massmoon*v^2/r

But I wonder what would hold Earth?

In fact the moon is not orbiting around Earth,but around a common center of gravity. That probably is too compliated for this thinking, after all, a massless cable almost 200,000 miles is a stretch. Pun intended.

To determine the tension in the cable, let's consider the scenario you described.

In this hypothetical situation, where the moon is held in its orbit by a long, mass-less cable attached to the center of the Earth, we can analyze the forces acting on the moon. The only force involved would be the tension in the cable since there is no gravitational interaction between the moon and the Earth.

The tension in the cable would act as the centripetal force required to keep the moon in its orbit. Centripetal force is defined as the force that keeps an object moving in a circular path, directed towards the center of the circle.

Using Newton's second law of motion, F = ma (force equals mass times acceleration), we can calculate the tension in the cable by equating it to the centripetal force.

The centripetal force is given by the formula Fc = (mv^2) / r, where m represents the mass of the moon, v is the velocity of the moon in its orbit, and r is the radius of the moon's orbit.

Since the moon's velocity and orbital radius depend on the specific scenario, let's assume an average velocity of 1 km/s and an average radius of 384,400 km (which is the mean distance from the Earth to the moon).

Now, let's plug these values into the formula:

Fc = (m * v^2) / r
= (m * (1,000 m/s)^2) / 384,400,000 m
= m / 384,400

Therefore, the tension in the cable would be equal to the mass of the moon divided by 384,400.

Keep in mind that this hypothetical scenario does not represent the actual forces and interactions responsible for the moon's orbit around the Earth. Gravitational attraction is the primary force that keeps celestial bodies in orbit.