Poorly stated and illustrated quiz problem. Your answer may be correct for the middle position, but not for the others; and I'm not sure about the middle.
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In vino veritas; in cervisia carmen; in aqua E. coli.
I am not sure I understand the problem so that I only try to explain what I understood.The geared sector rolls without sliding on the sun. Its COG is the black point at distance "e" from the sector center and a from the contact point to the sun in the middle position. The weight is considered as applied at the COG. When the sector "rolls" to a 45° position right or left due to the number of teeth ratio the rolling instantaneous center moves with 90° on the sun. The force gives a moment with respect to this center which has to be equal to the moment given by the sector weight in the new COG position. Since the COG is not known in proportion to the radii it is not possible to compute the force. It should be something somewhere which is either no clear in general or not clear to me.
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