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I'm searching for a way to express the angle of the arm of a balance instrument expressed in the weight that is present in one arm. For a two legged balance one could use the moment formula: moment = force * position but I am using a three legged balance with an angle of 120 degrees between each leg. Does such formula exist or could the moment formula be used in this situation?

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You need to know where the center of mass is for the three masses (on the three arms). This is of course just the position weighted mean - so if the scales are at position $\vec r_i$, and contain mass $m_i$, then the center of mass is at

$$\vec C = \frac{\sum{m_i \vec r_i}}{\sum{m_i}}$$

Note that we are using vector addition here - this is really just a 2D (or 3D) version of the formula you were using with the simple balance.

In order to convert this into an angle of deflection, you need to consider what restoring force appears when your balance becomes unbalanced - usually the center of mass is below the center of support so that deflection results in a restoring torque. But that depends on the details of the construction.

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This problem or rather question is very similar to a problem I solved in actuating a mirror's angle in two degrees of freedom (yaw and pitch) using a central fulcrum or pivot by which the mirror rotated, and three piezoelectric actuators that acted 120 degrees apart at the mirror's circular periphery. To transform three axis actuation to two axis orthogonal frame of reference requires something very similar to the Clarke transformation which is often used to control three phase motors with orthogonal axis controls, otherwise known as direct-quadrature or DQ control.

Anyway one way you might approach the problem, the way I approached the actuation problem, would be to draw a vector diagram in three space. That would be three, equal length vectors pointing outward from the origin (the point of rotation), 120 degrees apart. In my application I dealt with the force and displacement at each point of the actuators relative to the central pivot point. In your case the forces will be applied by each mass in the gravitational field. If there is no central restoring force in the balance (only the positional constraint of the pivot) then equilibrium will come about when the moments all balance one another. The equilibrium angle is then determined by the geometry of your balance design.

Considering the geometry, the the tip of each vector would terminate at each mass you are 'weighing'. In the practical design of the balance you will have to decide whether each weight will be fixed or else provided with a pivot joint at each location. But to start you can assume the force of the weight will always act downward from the point of attachment. From this geometrical arrangement, write the equations of the three vectors in x-y-z space relative to the yaw-pitch angles or direction cosines if you wish.

But there is one remaining equation that constrains, and ties the whole system together for a solution. The tip of each of the three vectors must remain co-planar to the balance's plane. The co-planar constraint can be enforced by requiring that the vector dot-cross product of the three vectors always equal zero.

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