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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 of 4.2 α–1 β–2 γ2 in terms of the new units.
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 of 4.2 α–1 β–2 γ2 in terms of the new units.
CBSE | Class 11 | NCERT | Physics | Units and Measurements
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 of 4.2 α–1 β–2 γ2 in terms of the new units.

Answer

Given that 1 calorie = 4.2 J = 4.2 kg m2 s–2

Expression for the standard formula for the conversion is as follows –

\[\]\[\frac{Unit(given)}{Unit(new)}={{\left( \frac{{{M}_{1}}}{{{M}_{2}}} \right)}^{x}}{{\left( \frac{{{L}_{1}}}{{{L}_{2}}} \right)}^{y}}{{\left( \frac{{{T}_{1}}}{{{T}_{2}}} \right)}^{z}}\]

Dimension of energy are –

\[\left[ {{M}^{1}}{{L}^{2}}{{T}^{-2}} \right]\]

On comparing this with the formula, we have x = 1, y = 2 and z =- 2

And M1 = 1 kg, L1 = 1m, T1 = 1s

and we are given that M2 = α kg, L2 = β m, T2 = γ s

\[\frac{Calorie}{Unit(new)}=4.2{{\left( \frac{1}{\alpha } \right)}^{1}}{{\left( \frac{1}{\beta } \right)}^{2}}{{\left( \frac{1}{\gamma } \right)}^{-2}}\]

Therefore, Calorie = 4.2 α–1 β–2 γ2

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