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

Given that ,
1 calorie =4.2 (1 kg ) `(1m^(2)) (1s^(-2))`
New unit of mass `=alpha kg`
Hence, in terms of the new unit `1kg =1/(alpha) =alpha^(-1)`
in terms of the new unit of length,
`1m =1/(beta) =beta^(-1) ` or `1m^(2)=beta^(-2)`
And, in terms of the new unit of time,
`1s=1/(gamma) =gamma^(-1)`
`1s^(2) =gamma^(-2)`
`1 s^(-2) =gamma^(2)`
`:.` calorie `=4.2 (1alpha^(-1)) (1 beta^(-2)) (1 y^(2)) =4.2 alpha^(-1) beta^(-2) gamma^(2)`