Suppose you system of units has one years as the unit of tiem , one light year as the unit of length and mass of earth as the unit of mass. How much 1 N be equal to the units of force in this system. Take 1 year = `3 xx 10^(5) s , ` 1 light year `= 9.5 xx 10^(15)` m and mass and earth `= 6 xx 10^(24)` kg .

Explanation

Let y denote year , 1 denotes light year and `m_c` mass of earth `n_1 U_1 = n_2 U_2`
`therefore n_2 = n_1 ((U_1)/(U_2))`
1 N is the unit of force and the dimensions of force are `MLT^(-2)`
`therefore n_2 = n_1 [(M_1)/(M_2)] [(L_1)/(L_2)] [(T_1)/(T_2)]^(-2)`
Here subscript 1 represents the system of units in whcih .1N. is the unit of force i.e. SI, and subscript 2 represents the new system.
`therefore n_2 = 1 xx [(1Kg)/(1m_e)] [(1m)/(1l)] [(1s)/(1y)]^(-2) = [(1kg)/(6 xx 10^(24)kg)] [(1m)/(9.5 xx 10^(15) m)] [(1s)/(3 xx 10^5 s)]^(-2)`
`= [((3xx 10^5)^2)/(6xx10^(24) xx 9.5 xx 10^(15))]=(9xx10^(10))/(6xx9.5xx10^(39))] = 1.58 xx 10^(-30)`
1N = 1.58 `xx 10^(-30)` units of force in the new system . `therefore 1 N = 1.58 xx 10^(-30) m_e 1 y^(-2)`