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,$n_2=6.67\times 10^{-11} [\dfrac{kg}{g}]^{-1}[\dfrac{m}{cm}]^3[\dfrac{s}{s}]^{-1}$
$=6.67 \times 10^{-11}[\dfrac{1000 g}{g}]^{-1}[\dfrac{100 cm}{cm}]^3.^{-2}$
$=6.67\times 10^{-11}^{-1}\times ^3\times 1$
$=6.67 \times 10^{-11}\times \dfrac{1}{1000}\times 100 \times 100 \times 100$
$=6.67 \times 10^{-8}$
$\therefore G=6.67\times 10^{-8} dyne\, cm^2g^{-2}$

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