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,,"New system"),(n_(1)=4.2,,n_(2)=?),(M_(1) = 1 kg,,M_(2) =alpha kg),(L_(1) = 1m ,,L_(2)= beta metre ),(T_(1)= 1 s ,,T _(2)=gamma "second" ):}`
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[(1kg)/(alpha kg )]^(1)[(1m)/(beta m )]^(2)[(1s)/(gamma s)]^(-2) =4.2 alpha ^(-1)beta^(-2)gamma^(2) `