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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? |
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Answer» Solution :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`. |
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