Physics

If the value of universal gravitational constant G is $$6.67 \times 10^{-11}Nm^{-2}/kg^2$$ in M.K.S. system, then determine its value in C.G.S. system using the dimensions.


SOLUTION
Given : $$G= 6.67 x 10^{-11} N m^{-2}/kg^2$$
To determine the value of G s (G.S. system formula used) 
$$n_2=n_1[\dfrac{M_1}{M_2}]^a[\dfrac{L_1}{L_2}]^b[\dfrac{T_1}{T_2}]^c$$
The dimensional formula of $$G = [M^{-1}L^3T^{-2}]$$ 
$$\therefore a=-1,b=3,c=-2$$
Suppose, 
$$n_2$$ dyne $$cm^2.g^{-2} = 6.67 x 10^{-11} N. m^2 kg^{-2}$$
$$\therefore n_1=6.67 \times 10^{-11}$$
Therefore,$$n_2=6.67\times 10^{-11} [\dfrac{kg}{g}]^{-1}[\dfrac{m}{cm}]^3[\dfrac{s}{s}]^{-1}$$
$$=6.67 \times 10^{-11}[\dfrac{1000 g}{g}]^{-1}[\dfrac{100 cm}{cm}]^3.[1]^{-2}$$
$$=6.67\times 10^{-11}[1000]^{-1}\times [100]^3\times 1$$
$$=6.67 \times 10^{-11}\times \dfrac{1}{1000}\times 100 \times 100 \times 100$$
$$=6.67 \times 10^{-8}$$
$$\therefore G=6.67\times 10^{-8} dyne\, cm^2g^{-2}$$
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