In a new system of units, value of speed of light `(c = 3xx 10^(8) m"/"s)`, Universal gravitational constant `(G= 6.67xx10^(-11))` and acceleration due to gravity `(g= 9.8 m"/"s^(2))` are found to be unity. What are the units of mass, length and time in this new system of units?

Explanation

Let `M_(1), L_(1)" and "T_(1)` represent fundamental units in MKS system and `M_(2), L_(2)" and "T_(2)` represent the same in new system of units. Let us assume that `M_(2)=x M_(1), L_(2)=y L_(1)" and "T_(2)= z T_(1)`.
Dimensional fourmula for speed of light is `LT^(-1)`. Let `n_(1)" and "n_(2)` be the values for speed of light in MKS and new system respectively.
`n_(1)u_(1)= n_(2)u_(2)`
`implies (n_1)[L_(1)T_(1)^(-1)](n_2)[L_(2)T_(2)^(-1)]`
`implies (n_1)=(n_2)[(L_2)/(L_1)][(T_(2)^(-1))/(T_(1)^(-1))]`
`implies (n_1)= (n_2)[(L_2)/(L_1)][(T_1)/(T_2)]`
`implies (3xx10^(8))=(1)(y)(1/z)`
`implies (y)/(z)=3xx10^(8)" ""............."(i)`
Dimensional formula for Gravitatioal constant is `M^(-1)L^(3)T^(-2)`. Let `n_(1)" and "n_(2)` be the values for gravitational constant in MKS and new system respectively.
`n_(1)u_(1)= n_(2)u_(2)`
`implies (n_1)[M_(1)^(-1)L_(1^(3))T_(1)^(-2)]=(n_2)[M_(2)^(-1)L_(2)^(3)T_(2)^(-2)]`
`implies (n_1)=(n_2)[(M_(2)^(-1))/(M_(1)^(-1))(L_(2)^(3))/(L_(1)^(3))(T_(2)^(-2))/(T_(1)^(-2))]`
`implies (n_1)= (n_2)[(M_1)/(M_2)][(L_2)/(L_1)]^(3)[(T_1)/(T_2)]^(2)`
`implies (6.67xx10^(-11))= (1)(1/x)(y^3)((1)/(z^2))`
`implies (y^3)/(xz^2)=6.67xx10^(-11)" "".........."(ii)`
Dimensional fourmula for acceleration due to gravity `LT^(-2)`. Let `n_(1)" and "n_(2)` be the values for acceleration due to gravity in MKS and new system respectively.
`n_(1)u_(1)= n_(2)u_(2)`
`implies (n_1)[L_(1)T_(1)^(-2)]=(n_2)[L_(2)T_(2)^(-2)]`
`implies (n_1)=(n_2)[(L_2)/(L_1)(T_(2)^(-2))/(T_(1)^(-2))]`
`implies (n_1)= (n_2)[(L_2)/(L_1)][(T_1)/(T_2)]^(2)`
`implies (9.8)= (1)(y)((1)/(z^2))`
`implies (y)/(z^2)=9.8" ""............"(iii)`
On dividing equation (iii) from equation (i) we get the following :
`z=(3xx10^(8))/(9.8)`
`implies z=3.06xx10^(7)" ""............"(iv)`
Substituting the value of z from equation (iv) in equation (i) we get the following :
`(y)/(3.06xx10(7))= 3xx10^(8)`
`implies y= 9.18xx10^(15)" ""............"(v)`
Now substituting the values of y and z from equation (iv) and (v) in equation (ii) we get the following :
`x=((9.18xx10^(15))^(3))/((6.67xx10^(-11))(3.06xx10^(7))^(2))`
`implies =(773.62xx10^(45))/((6.67xx10^(-11))(9.364xx10^(14)))`
`implies x= 12.4xx10^(42)" ""............"(vi)`
Now from results in equations (iv), (v) and (vi) we can describe the new system of units as follws :
Unit of mass `=12.4xx10^(42)kg`
Unit of length `=9.18xx10^(15)m`
Unit of time `=3.06xx10^(7)s`.