Balancing of a Single Rotating Mass By a Single Mass Rotating in the Same Plane

          Let us consider a disturbing mass (Extra mass) 'm1' attached to a shaft rotating at 'w rad/s as shown in Fig.,. Imagine, that mass 'm1' is rotating at a distance  'r1' (radius of rotation) (i.e., distance between the axis of rotation of the shaft and the centre of gravity of the mass).

We know that the centrifugal force exerted by any mass on the shaft,

                                  F_C = m .w². r

Then centrifugal force exerted by the mass 'm1' is 

                                 F_C1 = m1 .w². r1         . . . . . (i)

          We all know that centrifugal force always acts radially outwards and thus this centrifugal force produces bending moment on the shaft. In order to counteract the effect of this force, a balancing mass 'm2' may be attached in the same plane of rotation as that of disturbing mass 'm1' such that the centrifugal forces due to the two masses are equal and opposite.

Let r2= Radius of rotation of mass 

Centrifugal force due to mass 'm2',

                                 F_C2 = m2 .w². r2         . . . . . (ii)

Equating equations (i) and (ii)

                                m1 .w². r1 = m2 .w². r2

                                                  (or)

                                       m1 . r1 = m2 . r2

Note: 1. The centrifugal forces are proportional to the product of the masses and radius of rotation of respective masses, because 'w² ' is same for each mass.

2.