Units and Measurements — Physics STD 11 Science — Question
Maharashtra BoardEnglish MediumSTD 11 SciencePhysicsUnits and Measurements4 Marks
Question
Find the conversion factor between the $\text{S.I.}$ and the $\text{COES.}$ units of work using dimensional analysis.
✓
Answer
Conversion factor between units of same physical quantity:
$1.$ let $‘n \ ’$ be the conversion factor between the units of work.
$\therefore 1 J = n \ erg ……(1)$
$2.$Dimensions of work in $\text{S.l.}$ system are $\left[ L _1^2 M _1^{\prime} T _1^{-2}\right]$ and in $\text{CGS}$ system are $\left[ L _2^2 M _2^1 T_2^{-2}\right]$
$3.$From $(1),$
$1\left[L_1^2 M _1^1 T_1^{-2}\right]= n \left[ L _2^2 M _2^1 T_2^{-2}\right]$
$\therefore n =\left[\frac{ L _1^2 M _1^1 T_1^{-2}}{L_2^2 M _2^1 T_2^{-2}}\right]$
$ =\left[\frac{ L _1}{L_2}\right]^2\left[\frac{ M _1}{ M _2}\right]^1\left[\frac{ T _1}{T_2}\right]^{-2} .........(2)$
$4.$ By expressing $L , M$ and $T$ into its corresponding unit we have,
$n =\left[\frac{ m }{ \ cm }\right]^2\left[\frac{ \ kg }{ g }\right]^1\left[\frac{\text { second }}{\text { second }}\right]^{-2}........(3)$
$5.$Since, $1 m=100 \ cm$ and $1 \ kg=1000 g$, we have,
$n =\left(\frac{100 \ cm}{ \ cm }\right)^2\left(\frac{1000 g}{ g }\right)(1)^{-2}$
$n= 10^4 \times 10^3 \times 1 = 10^7$
Hence, the conversion factor, $n = 10^7$
$\therefore$ from equation $(1),$ we have,
$\therefore 1 J = 10^7 \ erg.$
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