JEE Main & Advanced Mathematics Statistics Centre of Gravity of a Compound Body and Remainder

(1) Centre of Gravity of a compound body :  Let \[{{G}_{1}}\],  \[{{G}_{2}}\] be the centres of gravity of the two parts of a body and let \[{{w}_{1}},{{w}_{2}}\] be their weights.

Let G be the centre of gravity of the whole body. Then, at G, acts the whole weight \[({{w}_{1}}+{{w}_{2}})\]of the body.

Join \[{{G}_{1}}{{G}_{2}}\]; then G must lie on \[{{G}_{1}}\]\[{{G}_{2}}\]. Let O be any fixed point on \[{{G}_{1}}{{G}_{2}}\]. Let \[O{{G}_{1}}={{x}_{1}},O{{G}_{2}}={{x}_{2}}\] and \[OG=\bar{x}\]. Taking moments about O, we have \[({{w}_{1}}+{{w}_{2}})\bar{x}={{w}_{1}}{{x}_{1}}+{{w}_{2}}{{x}_{2}}\]

or  \[\bar{x}=\frac{{{w}_{1}}{{x}_{1}}+{{w}_{2}}{{x}_{2}}}{{{w}_{1}}+{{w}_{2}}}\].

(2) Centre of gravity of the remainder : Let w be the weight of the whole body. Let a part B of the body of weight \[{{w}_{1}}\]be removed so that a part A of weight \[w-{{w}_{1}}\] is left behind.

Let G be the centre of gravity of whole body and \[{{G}_{1}}\], the C.G. of portion B which is removed. Let \[{{G}_{2}}\] be the C.G. of the remaining portion A. Let O be a point on \[{{G}_{1}}{{G}_{2}}\] and let it be regarded as origin. Let \[O{{G}_{1}}={{x}_{1}},OG=x\], \[O{{G}_{2}}={{x}_{2}}\] Taking moments about O,

\[(w-{{w}_{1}}){{x}_{2}}+{{w}_{1}}{{x}_{1}}=wx\]

or \[{{x}_{2}}=\frac{wx-{{w}_{1}}{{x}_{1}}}{w-{{w}_{1}}}\].