Question

convert 1 joule to ergs in dimensional analysis

Answer 1

$$\begin{gathered} \mathrm{Joule}\mathrm{is}\mathrm{the}\mathrm{unit}\mathrm{for}\mathrm{the}\mathrm{quantity}\mathrm{of}\mathrm{energy}. \\ \mathrm{Hence}, \\ \mathrm{Dimensional}\mathrm{formula}\mathrm{for}\mathrm{joule}\mathrm{is}\left[{\mathrm{M}}^1{\mathrm{L}}^2{\mathrm{T}}^{-2}\right] \\ \mathrm{Let} \\ {\mathrm{n}}_1\left(\mathrm{joule}\right)={\mathrm{n}}_2\left(\mathrm{erg}\right) \\ \mathrm{Hence}, \\ {\mathrm{n}}_1\left[{{\mathrm{M}}_1}^1{{\mathrm{L}}_1}^2{{\mathrm{T}}_1}^{-2}\right]={\mathrm{n}}_2\left[{{\mathrm{M}}_2}^1{{\mathrm{L}}_2}^2{{\mathrm{T}}_2}^{-2}\right]\mathrm{Here}1->\mathrm{SI}\mathrm{system}\mathrm{and}2->\mathrm{CGS}\mathrm{system} \\ {\mathrm{n}}_2={\mathrm{n}}_1\frac{\left[{{\mathrm{M}}_1}^1{{\mathrm{L}}_1}^2{{\mathrm{T}}_1}^{-2}\right]}{\left[{{\mathrm{M}}_2}^1{{\mathrm{L}}_2}^2{{\mathrm{T}}_2}^{-2}\right]} \\ ={\mathrm{n}}_1\times \left(\frac{{\mathrm{M}}_1}{{\mathrm{M}}_2}\right)\times {\left(\frac{{\mathrm{L}}_1}{{\mathrm{L}}_2}\right)}^2\times {\left(\frac{{\mathrm{T}}_1}{{\mathrm{T}}_2}\right)}^{-2} \\ =1\times \left(\frac{1\mathrm{kg}}{1\mathrm{g}}\right)\times {\left(\frac{1\mathrm{m}}{1\mathrm{cm}}\right)}^2\times {\left(\frac{1\mathrm{s}}{1\mathrm{s}}\right)}^{-2} \\ =1\times \left(\frac{1000\mathrm{g}}{1\mathrm{g}}\right)\times {\left(\frac{100\mathrm{cm}}{1\mathrm{cm}}\right)}^2\times {\left(\frac{1\mathrm{s}}{1\mathrm{s}}\right)}^{-2} \\ ={10}^7 \\ \mathrm{Therefore}, \\ {\mathrm{n}}_1\left(\mathrm{joule}\right)={\mathrm{n}}_2\left(\mathrm{erg}\right) \\ \mathbf{1}\mathbf{}\mathbf{joule}\mathbf{}\mathbf{=}\mathbf{}{\mathbf{10}}^{\mathbf{7}}\mathbf{}\mathbf{ergs} \end{gathered}$$