Engineering Mechanics

Chapter 29 : Balancing of Rotating Masses „„„„„ 589

m 3 , r 3 = Corresponding values for the balancing body C.

l 2 = Distance between the lines of action of the bodies A and B.

l 3 = Distance between the lines of action of the bodies A and C.

Now in order to satisfy the first condition,

m 1 r 1 = m 2 r 2 + m 3 r 3

and in order to satisfy the second condition,

m 1 r 1 l 1 = m 2 r 2 l 2 + m 3 r 3 l 3
Example 29.2. A 40 kg mass (A) mounted on an axle at a distance of 1 m is to be balanced
by two masses (B) and (C). The balancing masses are to be mounted in the planes 1 m and 2 m on
either sides of 40 kg mass at radii 1 m and 2 m respectively from the axis of rotation.

Find the magnitudes of the balancing masses.
Solution. Given : Mass of the body A (m 1 ) = 40 kg ; Distance between the centre of mass A and
the axis of rotation (r 1 ) = 1 m ; Distance between the lines of action of the bodies A and B (l 2 ) = 1 m ;
Distance between the lines of action of the bodies A and C (l 3 ) = 2 m ; Radius of the rotating body B (r 2 )
= 1 m and radius of the rotating body C (r 3 ) = 2 m

Let m 2 = Magnitude of the mass B in kg and

m 3 = Magnitude of the mass C in kg

We know that m 1 r 1 = m 2 r 2 + m 3 r 3

40 × 1 = m 2 × 1 + m 3 × 2

∴ m 2 + 2m 3 = 40 ...(i)

and m 2 r 2 l 2 = m 3 r 3 l 3

m 2 × 1 × 1 = m 3 × 2 × 2 = 4 m 3

∴ m 2 = 4 m 3 ...(ii)

Substituting the value of m 2 in equation (i),

4 m 3 + 2 m 3 = 40

∴ 3

40

6·67 kg

6

m == Ans.

and m 2 = 4 m 3 = 4 × 6·67 = 26·67 kg Ans.

EXERCISE 29.1

1. A body of mass 30 kg is attached to a shaft rotating at 300 r.p.m. at a distance of 500 mm
   from its axis. The body is to be balanced by mass, which has to be attached at a distance
   of 300 mm from the axis of the shaft. Find the magnitude of the balancing mass.
   (Ans. 50 kg)


2. A body of mass 10 kg is attached to a rotating shaft at a radius of 500 mm from its axis of
   rotation. It is to be balanced by two bodies with their centres of gravity in the same plane
   in such a way that one of the mass is 200 mm from 10 kg mass and the other 300 mm on
   the opposite side. Find the masses of the balancing bodies, if their centres of gravity are at
   a distance of 400 mm from the axis of the rotating shaft. (Ans. 7·5 kg)