A calorie is a unit of heat (energy in transit) and it equals about 4.2 J where 1J = 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.
A calorie has a magnitude 4.2 α⁻¹ β⁻² γ² in terms of the new units.
Explanation
This problem tests the understanding of unit conversion using dimensional analysis. The key concept is that the physical quantity (energy) remains the same regardless of the system of units used. The dimensional formula of energy is [M L² T⁻²]. When converting from SI units to a new system, we use the relationship between the old and new units for mass, length, and time. The numerical value changes inversely with the size of the unit - larger units lead to smaller numerical values and vice versa.
Solution Steps
Step 1: Write the given relation for calorie in SI units. 1 calorie = 4.2 J = 4.2 kg m² s⁻²
Step 2: Express the relationship between SI units and new units. Given: 1 unit of mass = α kg, so 1 kg = (1/α) new mass units 1 unit of length = β m, so 1 m = (1/β) new length units 1 unit of time = γ s, so 1 s = (1/γ) new time units
Step 3: Substitute these values in the expression for calorie. 1 calorie = 4.2 × (1/α) × (1/β)² × (1/γ)⁻² new energy units
Step 4: Simplify the expression. 1 calorie = 4.2 × α⁻¹ × β⁻² × γ² new energy units Therefore, a calorie has a magnitude 4.2 α⁻¹ β⁻² γ² in terms of the new units.