If the moon did not rotate at the same rate that it revolved, what would happen to the gravity of the earth?

The gravity, which is an acceleration to the center of the earth, will be the same.

Explanation:

The gravity on earth depends only on the masses and distance, between two objects. We can see it in the gravitational force equation.  

Now if we put a man, with mass m, on the surface of the earth, with mass M, the distance from the center of mass and the man will be R (earth radius). Knowing that F = m*a, we can find the accelerations due to this mass M and this value will be 9.81 m/s².  

On the other hand, the moon has a gravity value and is less than the earth, because its mass, and affects the water sea due to the gravitational force between earth and moon. If the moon changes the rate of its rotate it changes probably the distance between them, let's recall they must conserve angular momentum, but the gravity won't be affected.

Therefore, the gravity, which is an acceleration to the center of the earth, will be the same.

I hope it helps you!

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explanation: