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If we have an empty merry-go-round where there is no friction and can be treated as a uniform disk. If there is a constant force of F, then its angular acceleration is given by the equation: $$\frac{\tau_{net}}{\frac{1}{2}MR^2} = \alpha$$ However, if we add a person onto this merry-go-round then the angular acceleration is given by the equation $$\frac{\tau_{net}}{\frac{1}{2}MR^2 + mr^2} = \alpha$$. Why don't we consider the force of gravity that the person exerts on the merry-go-round due to their gravity and that torque, whe can we consider it as the same torque for both scenarios?

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Here we are considering the rotation of a disk about its axis of symmetry. The other degrees of rotation are restricted. So any torque applied to this disk must point along the axis of symmetry to have any affect on the rotation. In the example you gave, the angular acceleration of the disk changes because its moment of inertia changes with the addition of a person. There is a torque associated to the weight of that person, but it is directed in the plane of the disk, not along its axis of symmetry.

If you remove the constraint on the degrees of rotation, you would have to consider the torque from the weight of the person. For example, say you balanced the merry-go-round on the head of a pin. Then you would have to consider the additional torque from gravity in your calculations.

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Because you are only considering the horizontal situation.

In vertical situation, the torque exerted by gravity is equal to the torque exerted by normal force.

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