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When taking moments around B, with AB as a road and AC a string, why don't we consider Tcos(alpha) when balancing anti clockwise and clockwise forces? (the diagram is in equilibrium)

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That component of the tension has a radial direction and therefore exerts no torque.
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You can ignore forces that pass through the axis you are taking the torque about.

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Think about a door. If you had the door open and pushed onto the door directly towards the hinges it wouldn't open/close any further. Torque will not be affected by the forces that go towards/away from the axis.
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I know this isn't techincally geometry but there was no mechanics flair sorry Reddit mods

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