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Dimensions of physical quantities

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Question

In a hypothetical set of units, mass, acceleration and frequency are taken as fundamental units. If unit of mass is 100 g, unit of acceleration is 2 m/s² and unit of frequency is 4 sec^-1 in the new system of units then find the value of one joule in this system.

Moderate

Solution

Energy $\propto m^{x} a^{y} f^{z}$
$\begin{array}{l} [{ML}^{2} T^{- 2}] = [M]^{x} {[{LT}^{- 2}]}^{y} {[T^{- 1}]}^{z} \\ 1 = x \\ 2 = y \\ - 2 = - 2 y - z \Rightarrow z = - 2 \end{array}$
$Energy \propto \frac{{ma}^{2}}{f^{2}}$
$\begin{array}{l} 1 .0 J = {xJ}^{'} \\ x = (1 .0) \frac{J}{J^{'}} = (1 .0) {(\frac{M_{1}}{M_{2}})}^{1} {(\frac{a_{1}}{a_{2}})}^{2} {(\frac{f_{1}}{f_{2}})}^{- 2} \\ = (1 .0) (\frac{1 kg}{100 g}) {(\frac{1 m / s^{2}}{2 m / s^{2}})}^{2} {(\frac{1 \sec^{- 1}}{4 \sec^{- 1}})}^{- 2} \\ = (1 .0) (10) (\frac{1}{4}) (16) = 40 \end{array}$

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