Question

A calorie is a unit of heat (energy in transit) and it equals about 4.2 J where 1J = 1 \(\text {kg} \;\text m^2\; \text s^{-2}.\) Suppose we employ a system of units in which the unit of mass equals \(\alpha\) kg, the unit of length equals \(\beta\) m, the unit of time is \(\gamma\) s. Show that a calorie has a magnitude 4.2 \(\alpha^{-1}\; \beta^{-2}\; \gamma^{2}\) in terms of the new units.

Answer 1

Given:

1 calorie = 4.2 J

1 joule is defined as:

1 J = 1 kg m² s⁻²

New system of units:

• Unit of mass = α kg
• Unit of length = β m
• Unit of time = γ s

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Step 1: Express SI units in terms of new units

From the definitions:

1 kg = (1/α) × (new unit of mass)

1 m = (1/β) × (new unit of length)

1 s = (1/γ) × (new unit of time)

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Step 2: Express 1 joule in the new units

1 J = 1 kg m² s⁻²

Substitute the above relations:

1 J = (1/α) × (1/β)² × (1/γ)⁻²

1 J = α⁻¹ β⁻² γ² (new unit of energy)

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Step 3: Express one calorie in the new units

1 calorie = 4.2 J

Therefore:

1 calorie = 4.2 × α⁻¹ β⁻² γ²

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Final Answer:

In the new system of units, the magnitude of a calorie is:
4.2 α⁻¹ β⁻² γ²