What would happen to our moon if the Earth had the shape of a doughnut ?

Most likely, it will be pulled towards the hole, and it will swing up and down in the center of the planet. Or it may be affected by the gravity of the outer edges of the donut, and it will create an eightfold orbit around the Earth.
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Aurora Borealis photobombs the Milky Way
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Kufstein, Tyrol, Austria Hungary around 1900, colourized
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28-year-old Jenny Joseph modeling for the Columbia Pictures logo. She had never modeled beforehand and never modeled after this.
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Making the flame of Liberty, 1876
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If the Moon were a giant mirror.
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Opposites attract
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French Military Training, c. 1896-1899
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Amsterdam
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Suspended unipolar motor

The principle of operation is based on the force generated by the interaction of an electric current with a magnetic field (Lorentz force, also known as Laplace force).
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