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

`n_(2)=n_(1)[M_(1)/M_(2)]^(a)[L_(1)/L_(2)]^(b)[T_(1)/T_(2)]^(c)` ...(i)
Given 1 cal `=4.2 J=4.2 kg m^(2)s^(-2)`
Therefore, `a=1, b=2, c=-2`,
Also,
`{:(n_(1)=4.2,|n_(2)=?,),(M_(1)=1 kg,M_(2)=alpha kg,),(L_(1)=1 m,L_(2)=beta" metre",),(T_(1)= 1s,T_(2)= gamma" second",):}`
Substituting these values in eqn. (i) we get
`n_(2)=4.2[(1 kg)/(alpha kg)]^(1)[(1m)/(beta m)]^(2)[(1s)/(gamma s)]^(-2)`
`=4.2 alpha^(-1) beta^(-2) gamma^(2)`