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If a solid sphere is moving vertically down with a velocity $v$ and spinning with an angular velocity $\omega$, what is the angular momentum of the sphere about an axis through the sphere which is at a distance $ \frac{R}{2}$ vertically below the center of mass of the sphere ?

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If the sphere is not spinning, and the axis about which you are taking angular momentum stays through the same position in the sphere, then the relative velocity of the sphere to the axis will always be zero regardless of translational motion. Therefore, the angular momentum will be zero.

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