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Centre of Gravity and Centre of Mass

Refer Fig. 6.2(a) the body of mass m is suspended by an inextensible cord from point a. The weight W = mg will be acting vertically downward. We draw the vertically downward direction by dotted line from point a. Then consecutively the body is suspended from points band c respectively [ Fig. 6.2(b) and Fig. 6.2(c)]. All vertical dotted lines meet at a point ‘G’ called Centre of Gravity (C.G.). The center of gravity is a fixed point on a body.

 

To determine C.G. mathematically, we refer to Fig. 6.3(a) and 6.3(b).

Taking moment of dW about y axis, we have, dM = x dW

 

or                      M = ʃ x dW for all the distributed forces

 

Taking moment of the resultant (W) of all dW’s about y axis, we have,

M1  = xW

By Varignon’s theorem (Principle of moments) we can write

M1 =M

x W = ʃ x dW

x= x dW / W

 

Similarly we can write the expression for y and z as follows

 

Y = ʃ y dW /W , z =  ʃ z dW /W

 

Thus, the coordinate of G, G (x, y,z) is obtained.

 

x = ʃ x dW /W , y = ʃ y dW /W ,z = ʃ z dW/W

 

We have                      W = mg, so equation (6.1) can be rewritten as

 

x = ʃ x dm /m , y = ʃ y dm /m ,z = ʃ z dm/m

 

Now we express          m= Pv ,           dm = pdV

 

x = ʃ xp dV / ʃ p dV,  y = ʃ y dV / ʃ p dV , z = ʃ z dV / ʃ p dV

 

Equation (6.2) may be expressed in vector form (Refer Fig. 6.3(b))

Equations (6.2), (6.3) and (6.4) contain no reference of ‘g’ (gravitational effect). Therefore, we get a unique point (X, y, Z) which is known as the centre of mass. The centre of mass coincides with the centre of gravity as long as g is constant.