2.3 A calorie is a unit of heat (energy in transit) and it equals about 4.2 J where 1 J = 1 kgm2s–2. Suppose we employ a system of units in which the unit of mass equals α kg, the unit of length equals β m, the unit of time is γ s. Show that a calorie has a magnitude of 4.2 α–1β–2γ2 in terms of the new units.

 
 
Hint: We have to consider the dimensional formulas of the quantities given and use the values given in the question.
Step 1: Conversion in the new unit as:
1m'=βm1kg'=αkg1s'=γs1m=1βm'
1kg=1αkg'1s=1γs'
Step 2:
1J=1kg×(1m)2×(1s)-2
=1αkg'×1βm'2×1γs'-2
 
=γ2α-1β-2kg'm'2s'-21cal=4.2α-1β-2γ2.