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Say, in 2 dimensions, that we have a rectangular mass resting on two legs. Then each leg supports half its weight. Now, say one of those legs is instead a string hanging from the ceiling. How is weight distributed now? How is torque factored in? What if the rectangular mass is not parallel to the floor but offset by 45 degrees — does this change anything?

I am trying to answer these questions to understand the force diagram of a sailboat trapeze (https://www.sailingscuttlebutt.com/wp-content/uploads/2016/02/trapeze.jpg) so I can write a formula for torque on the boat given the person's weight, angle relative to the boat, and height. The person stands on the side of the boat, and leans back while connected to a wire attached to the mast, causing the boat to rotate over. If anyone could help with this as well, it would be much appreciated, but if you can only answer the first paragraph, please only worry about that.

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  • $\begingroup$ “Then each leg supports half its weight“ ... only if the legs are the same distance from the object’s centre of mass. $\endgroup$
    – gandalf61
    Commented Sep 19, 2021 at 6:32

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For a mass supported with two forces F1 and F2, the vertical components must balance the weight of the mass: mg + F1,z + F2,z = 0. If the mass does not spin, then the torques must be balanced as well. This is why gandalf comments that the distance from the centre of mass is important. If you imagine one leg is directly below the center of mass providing 0 torque, if the other leg is not directly below the center of mass but contributes any force and torque, it would cause the mass to spin.

For the case of the boat, regardless of the wires/legs holding the man, his torque on the boat center of mass (COM) will be $\vec{r} x m\vec{g}$ where $\vec{g}$ is the direction of gravity and $\vec{r}$ is the displacement vector from the COM of the boat to the man.

The sum of the wires/leg forces will balance $\vec{g}$.

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  • $\begingroup$ Thank you. One quick question—what is x in the equation in the second paragraph? $\endgroup$
    – Gamma05772
    Commented Sep 19, 2021 at 13:59
  • $\begingroup$ cross product x $\endgroup$
    – Alwin
    Commented Sep 19, 2021 at 19:41

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