Balancing of Several Masses in Different Transverse Planes:

Now we will consider a case, where there are various masses in different planes.

To start with, we shall name the planes in which masses revolve like A, B, C and D. The masses rotating in these planes are W_a, W_b, W_c and W_d respectively. The radii at which centres of gravity lie through axis of rotation in these planes are r_a, r_b, r_c and r_d respectively.

The angular separation among masses beginning from A and B are α, β and γ. The balancing masses shall be located in two planes L and M which are among A and B and among C and D, respectively. Figure 5 depicts the system. Distance among planes L and any of planes A, B, C and D is mentioned by l with suitable suffix. The similar distances for plane M is mentioned by m.

The method clearly is same as utilized in Section 19.3.1 in which balance masses in two planes L and M were found. So we have to repeat the process for mass in plane A and balancing mass in planes L and M. Therefore we proceed in steps of planes A, B, C and D. A table of the type shown below as Table 1 shall be helpful. Rows shall be devoted to planes in which masses revolve that could be known or unknown.

For a mass of weight W, revolving at radius r, the force is proportional to W r as ω² /g is a constant. Two planes L & M are selected which are respectively at distances of l & m from the plane of revolving mass and balancing masses are located in planes L and M as given by Equation (6) and (7). Therefore the columns of the table shall describe plane (A, B, C, etc.) weight W; radius r; force W r; distances l and m; balancing forces in L and M.

                  Table 1: Calculation of Balancing Masses in Two Planes

In last two columns the sign of the forces shall be decided by observation. Still as a rule if a force is in the similar direction as the disturbing force, then it shall be positive and if in the opposite direction it shall be -ve. After the balancing forces have been measured in planes L & M which shall be parallel to forces in planes A and B, etc. they are joints to provide a single resultant. Their inclination to force in plane A might be calculated.