# Is my angular momentum equal to that of the Earth?

Yes, this question sounds silly. However someone who ought to know better insists (with insults) that a person standing on a planet has the same angular momentum as the planet. They certainly have the same angular velocity, but then to have the same angular momentum, wouldn't I have to have the same mass as the Earth? What am I missing?

• you are not missing anything, you can insult that guy back.
– user178659
Jun 17, 2021 at 21:40
• Actually, being part of the Earth, your angular momentum is one tiny part of the angular momentum of the Earth. Jun 18, 2021 at 14:19

Somebody who ought to know better is wrong, and you're almost right.

$$L=\omega I$$

Where I, the moment of inertia, is the mass times the square of the distance from the center of rotation to the distance at which half of the mass is farther from the center than you. For a solid sphere of radius R, mass M, which approximates Earth if we assume constant density,

$$I=\frac{2}5 MR^2$$ (Derivation: http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html#mi )

While for a point mass

$$I=MR^2$$

So if you were at the equator you would only have to mass 2/5 as much as the Earth to have the same moment of inertia and thus the same angular momentum, up to near infinite mass if you were standing at one of the poles.

• Constant density isn't a great approximation for the Earth, and in reality the factor is around 0.33 rather than 0.4. Jun 17, 2021 at 22:29

Angular momentum is $$L = I \omega$$ where $$I$$ is the moment of inertia and $$\omega$$ is the angular velocity. If you are on the earth then your angular velocity is equal to the angular velocity of the earth. But to have the same angular momentum would require having the same a moment of inertia.

If you have the same moment of inertia as the earth then you really should see a doctor, but they are probably all dead because you ate the whole planet.

• Fun, but not correct as you would only need $2/5$ of the earth's mass to have the same moment of inertia on its surface. (assuming a homogeneous mass distribution inside the Earth) (assuming you are on the equator) (assuming the axis of rotation nor its angular velocity are affected by your mass) (nonetheless, absolutely go see a doctor) Jun 18, 2021 at 10:04