Ncert Exercises & Solutions

A calorie is a unit of heat (energy in transit) and it equals about \(4.2~\text J\) where \(1~\text J = 1~\mathrm{kgm^{2}s^{–2}}.\) Suppose we employ a system of units in which the unit of mass equals \(\alpha~\text{kg},\) the unit of length equals \(\beta~\text{m}\), the unit of time is \(\gamma~\text{s}.\) What is the magnitude of a calorie in terms of the new units?
1. \(4.2\alpha^{–1}\beta^{–1}\gamma^2\)
2. \(4.2\alpha^{–2}\beta^{–2}\gamma^2\)
3. \(4.2\alpha^{–1}\beta^{–2}\gamma^2\)
4. \(4.2\alpha^{–1}\beta^{–2}\gamma^1\)

Hint: We have to consider the dimensional formulas of the quantities given and use the values given in the question.

Step 1: Conversion in the new unit as:

\begin{array}{l} 1 m' = β m \\ 1 kg' = α kg \\ 1 s' = γ s \\ 1 m = \frac{1}{β} m' \end{array}

\begin{array}{l} 1 kg = \frac{1}{α} kg' \\ 1 s = \frac{1}{γ} s' \end{array}

Step 2:

1 J = 1 kg \times (1 m)^{2} \times (1 s)^{- 2}

= \frac{1}{α} kg' \times {(\frac{1}{β} m')}^{2} \times {(\frac{1}{γ} s')}^{- 2}

\begin{array}{l} = γ^{2} α^{- 1} β^{- 2} kg' m'^{2} {(s')}^{- 2} \\ 1 cal = 4.2 α^{- 1} β^{- 2} γ^{2} . \end{array}

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