Question

In a new system of units the units of force, energy and velocity are 10N, 0.1J and 100m/s. What are the units of length, time and mass in this system?

Answer 1

$$\begin{gathered} \mathrm{Force}=10\mathrm{N},\mathrm{Energy}=0.1\mathrm{J}\mathrm{and}\mathrm{velocity}=100\mathrm{m}/\mathrm{s} \\ \mathrm{Suppose}{\mathrm{M}}_1,{\mathrm{L}}_1\mathrm{and}{\mathrm{T}}_1\mathrm{are}\mathrm{the}\mathrm{units}\mathrm{of}\mathrm{mass},\mathrm{length}\mathrm{and}\mathrm{time}\mathrm{in}\mathrm{SI}\mathrm{and}{\mathrm{M}}_2,{\mathrm{L}}_2\mathrm{and}{\mathrm{T}}_2\mathrm{the}\mathrm{corresponding}\mathrm{units}\mathrm{in}\mathrm{new}\mathrm{system}. \\ \mathrm{The}\mathrm{dimensional}\mathrm{formula}\mathrm{for}\mathrm{force}\mathrm{is}\mathrm{given}\mathrm{by}, \\ \left[{\mathrm{M}}^1{\mathrm{L}}^1{\mathrm{T}}^{-2}\right] \\ \mathrm{Thus},\mathrm{the}\mathrm{conversion}\mathrm{formula}\mathrm{for}\mathrm{force}\mathrm{becomes}, \\ {\mathrm{n}}_2={\mathrm{n}}_1\left[\frac{{\mathrm{M}}_1}{{\mathrm{M}}_2}{]}^1\right[\frac{{\mathrm{L}}_1}{{\mathrm{L}}_2}{]}^1[\frac{{\mathrm{T}}_1}{{\mathrm{T}}_1}{]}^{-2} \\ \mathrm{Here}, \\ {\mathrm{n}}_1=10\mathrm{N}\mathrm{and}{\mathrm{n}}_2=1, \\ \mathrm{So},1=10[\frac{{\mathrm{M}}_1}{{\mathrm{M}}_2}{]}^1\left[\frac{{\mathrm{L}}_1}{{\mathrm{L}}_2}{]}^1\right[\frac{{\mathrm{T}}_1}{{\mathrm{T}}_1}{]}^{-2}----\left(\mathrm{i}\right) \\ \mathrm{The}\mathrm{dimensional}\mathrm{formula}\mathrm{for}\mathrm{energy}\mathrm{is}\left[{\mathrm{M}}^1{\mathrm{L}}^2{\mathrm{T}}^{-2}\right] \\ {\mathrm{n}}_2={\mathrm{n}}_1\left[\frac{{\mathrm{M}}_1}{{\mathrm{M}}_2}{]}^1\right[\frac{{\mathrm{L}}_1}{{\mathrm{L}}_2}{]}^2[\frac{{\mathrm{T}}_1}{{\mathrm{T}}_1}{]}^{-2} \\ \mathrm{Here},{\mathrm{n}}_1=0.1\mathrm{J}\mathrm{and}{\mathrm{n}}_2=1 \\ \mathrm{So},1=0.1[\frac{{\mathrm{M}}_1}{{\mathrm{M}}_2}{]}^1\left[\frac{{\mathrm{L}}_1}{{\mathrm{L}}_2}{]}^2\right[\frac{{\mathrm{T}}_1}{{\mathrm{T}}_2}{]}^{-2}----\left(\mathrm{ii}\right) \\ \mathrm{Similarly}\mathrm{the}\mathrm{dimensional}\mathrm{formula}\mathrm{for}\mathrm{velocity}\mathrm{is}\left[{\mathrm{M}}^0{\mathrm{L}}^1{\mathrm{T}}^{-1}\right] \\ {\mathrm{n}}_2={\mathrm{n}}_1\left[\frac{{\mathrm{M}}_1}{{\mathrm{M}}_2}{]}^0\right[\frac{{\mathrm{L}}_1}{{\mathrm{L}}_2}{]}^1[\frac{{\mathrm{T}}_1}{{\mathrm{T}}_2}{]}^{-1} \\ \mathrm{Here},{\mathrm{n}}_1=100\mathrm{m}/\mathrm{s},{\mathrm{n}}_2=1 \\ \mathrm{So},1=100[\frac{{\mathrm{M}}_1}{{\mathrm{M}}_2}{]}^0\left[\frac{{\mathrm{L}}_1}{{\mathrm{L}}_2}{]}^1\right[\frac{{\mathrm{T}}_1}{{\mathrm{T}}_2}{]}^{-1}---\left(\mathrm{iii}\right) \\ \mathrm{Dividing}\left(\mathrm{ii}\right)\mathrm{by}\left(\mathrm{i}\right), \\ 1=\frac{0.1}{10}\left[\frac{{\mathrm{L}}_1}{{\mathrm{L}}_2}\right] \\ \mathrm{or}\left[\frac{{\mathrm{L}}_1}{{\mathrm{L}}_2}\right]=\frac{10}{0.1}=100 \\ \mathrm{or}{\mathrm{L}}_2=\frac{{\mathrm{L}}_1}{100}=\frac{1}{100}=\mathbf{0}\mathbf{.}\mathbf{01}\mathbf{}\mathbf{m} \\ \mathrm{Substituting}\left[\frac{{\mathrm{L}}_1}{{\mathrm{L}}_2}\right]=100\mathrm{in}\left(\mathrm{iii}\right) \\ 1=100\times 100\left[\frac{{\mathrm{T}}_1}{{\mathrm{T}}_2}{]}^{-1} \\ \right[\frac{{\mathrm{T}}_1}{{\mathrm{T}}_2}]=10000 \\ \mathrm{or}{\mathrm{T}}_2=\frac{1}{10000}=\mathbf{0}\mathbf{.}\mathbf{0001}\mathbf{}\mathbf{s} \\ \mathrm{Substituting}\mathrm{value}\mathrm{of}[\frac{{\mathrm{L}}_1}{{\mathrm{L}}_2}]\mathrm{and}[\frac{{\mathrm{T}}_1}{{\mathrm{T}}_2}]\mathrm{in}(\mathrm{i}) \\ 1=10[\frac{{\mathrm{M}}_1}{{\mathrm{M}}_2}{]}^1\left[\frac{{\mathrm{L}}_1}{{\mathrm{L}}_2}{]}^1\right[\frac{{\mathrm{T}}_1}{{\mathrm{T}}_2}{]}^{-2} \\ 1=10\times [\frac{{\mathrm{M}}_1}{{\mathrm{M}}_2}{]}^1\times 100\times [10000{]}^{-2} \\ \frac{\left[10000\right]\times \left[10000\right]}{10\times 100}=\left[\frac{{\mathrm{M}}_1}{{\mathrm{M}}_2}\right] \\ 100000=\left[\frac{{\mathrm{M}}_1}{{\mathrm{M}}_2}\right] \\ \mathrm{or}{\mathrm{M}}_2=\frac{1}{100000}=\mathbf{0}\mathbf{.}\mathbf{00001}\mathbf{}\mathbf{kg} \end{gathered}$$