A calorie is a unit of heat or energy and it equals about 4.2 J where 1J = 1 kg m2s–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 γs. Show that a calorie has a magnitude 4.2 α–1 β–2 γ2 in terms of the new units.

A calorie is a unit of heat or energy and it equals about 4.2 J where 1J = 1 kg m²s^–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 γs. Show that a calorie has a magnitude 4.2 α^–1 β^–2 γ² in terms of the new units.

Solutions:

 Given that,

1 calorie = 4.2 j

1J = 1 kg m²s^–2

Plug the unit of 1 j we get

1 calorie =4.2  kg m²s^–2  = 4.2 × (1 kg) × (1 m²) × (1 s^–2)     …(1)

Given that

Unit of mass equals α kg  => 1 kg mass  = 1/α  mass

Unit of length equals β m => 1 m length = 1/ β length

Unit of time is γs                 => 1 s time  = 1/ γ  time

Plug above values in equation (1), we get

 New unit of mass = α kg

=4.2 × (1 kg) × (1 m²) × (1 s^–2)

= 4.2 × (1/ α ) × (1/ β ) ² × (1/ γ)^–2

= 4.2 × ( α^–1  ) × (β^–1  ) ² × ( γ^–1) ^–2

=4.2 α^–1 β^–2γ²

Hence proved