Torque due to earth's gravity on moon

[frostking]

Homework Statement

What is the magnitude of the torque caused by the force of gravity on the moon by the earth? Assume both are spheres of uniform density, the axis of rotation passes through the center of the Earth perpendicular to the plane of the moon's orbit. Earth's mass 5.98 x 10^24 kg, 7.36 x 10^22 kg for moon radius of moon's circular orbit is 3.84 x 10^8

Homework Equations

Torque = r x Fof gravity Fof gravity = GMm/r^2

The Attempt at a Solution

I know I am wrong now but I multiplied the distance between Earth and moon by forge of gravity of earth. I know from the answer key that the Earth's gravity does not exert any torque on the moon. However, I do not understand why. The Earth's center of mass is perpendicular to moon is the key I am sure. Is it because the force of Earth's g on moon is along the same axis of moon's and therefore no torque occurs along the axis of the pivot point? Because I definitely know that the Earth's gravity exerts torque on things on our planet like if a wrench breaks loose a nut from a screw gravity torque is what is acting Right?! Thanks very much for your time and effort. Frostking

 

[Delphi51]

Torque is "twisting force". You need torque to turn a bolt.
But the Earth's gravity pulls equally on both sides of the moon, so it does not twist it.

At least that is the obvious answer. There may in fact be some small twisting force involved because the moon rotates so the same part of it always faces the Earth. That must have been somehow caused by the Earth's gravity over millions of years.

 

[kerimek]

Your attempt at a solution is incorrect. The torque due to Earth's gravity on the moon is actually zero. This is because the force of gravity between two objects (in this case, Earth and the moon) is always directed along the line connecting their centers of mass. Since the axis of rotation passes through the center of the Earth and is perpendicular to the moon's orbit, the force of gravity between the two objects does not create any torque.

To better understand this, imagine a seesaw with two people on either end. If one person is significantly heavier than the other, they will exert a greater force on the seesaw, causing it to tilt towards them. However, if the seesaw is perfectly balanced and the two people are equally distant from the pivot point, there will be no torque and the seesaw will remain level. This is similar to the relationship between Earth and the moon - the force of gravity exists, but it does not create any torque because the two objects are balanced and equally distant from the axis of rotation.

In regards to your example of a wrench breaking loose a nut from a screw, this is not due to the force of gravity, but rather the force of the wrench itself. When you apply a force to the wrench, it creates a torque on the nut, causing it to loosen. This is because the force applied by the wrench is not directed along the axis of rotation, unlike the force of gravity between Earth and the moon.

I hope this helps to clarify the concept of torque and its application to the relationship between Earth and the moon.