A calorie is a unit of heat or energy and it equals about `4.2 J, where 1 J = 1 kg m^(2) s^(-2)`. Suppose we employ a system of units in which the unit of mass equals `alpha kg`, the unit of length equals is `beta m` , the unit of time is `gamma s`. Show tthat a calorie has a magnitude `4.2 alpha^(-1) beta^(-1) gamma^(2)` in terms of the new units.

Explanation

`1 cal = 4.2 kg m^(2) s^(-2)`
`|{:(SI" system",, "NEW system"),(M_(1)=1kg,,M_(2)=alpha kg),(L_(1)=1m,,L_(2)=betam),(T_(1)=1s,,T_(2)=gamma s),(n_(1)=4,,n_(2)=?):}|`
dimensional formula of energy is `[ML^(2)T^(-2)]`
Comparing with `[M^(a) L^(b)T^(c)]` , we find that `a = 1 , b = 2 , c= -2`.
Now, `n_(2) = n_(1) [(M_(1))/(M_(2))]^(a) [[(L_(1))/(L_(2))]^(b)[(T_(1))/(T_(2))]^(c)`
`= 4.2 [(1 kg)/( alpha kg )]^(1) [ ( 1m)/( beta m) ]^(2) [ ( 1s) /( gamma s)]^(-2) = 4.2 alpha ^(-1) beta^(-2) gamma^(2)`