Exercises
A calorie is a unit of heat (energy in transit) and it equals about 4.2 J where 1 J = 1 kg m² s⁻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 4.2 α⁻¹ β⁻² γ² in terms of the new units.
Energy has dimensions [M L² T⁻²].
In the new system: unit of mass = α kg, unit of length = β m, unit of time = γ s.
The new unit of energy = (α kg)(β m)²(γ s)⁻² = α β² γ⁻² joules
Number of new energy units in 4.2 J:
n = 4.2 J / (α β² γ⁻² J per unit)
= 4.2 / (α β² γ⁻²)
= 4.2 × α⁻¹ × β⁻² × γ²
Therefore, 1 calorie = 4.2 α⁻¹ β⁻² γ² in the new system of units. (Proved)
Answer: 1 calorie = 4.2 α⁻¹ β⁻² γ² new units (derived from dimensional analysis of [M L² T⁻²])