Obtain the relation between the units of some physical quantity in two different systems of units. Obtain the relation between the $MKS$ and $CGS$ unit of work.
The unit of work in $MKS$ system is Joule and that in $CGS$ system is erg. The relation between Joule and erg can be obtained as follows.
Dimensional formula for work, $\mathrm{W}=\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-2}$
in $MKS$ system = in $CGS$ system
$ \mathrm{M}(\mathrm{kg})=10^{3} \mathrm{M}(\mathrm{gm}) $
$ \mathrm{L}(\mathrm{m}) =10^{2} \mathrm{~L}(\mathrm{~cm}) $
$\mathrm{T}(\mathrm{s}) =10^{0} \mathrm{~T}(\mathrm{~s}) $
$ \mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-2} =\left(10^{3} \mathrm{M}\right)^{1}(10^{2} \mathrm{~L})^{2}\left(10^{0} \mathrm{~T}\right)^{-2}$
$ =10^{3} \times 10^{4} \mathrm{M}^{1} \mathrm{~L}^{2}\mathrm{~T}^{-2}$
$ =10^{7} \mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-2}$
So, $MKS$ unit of work $=10^{7}$ CGS unit of work $\therefore 1 \mathrm{~J}=10^{7} \mathrm{erg}$
Dimensional formula for work, $\mathrm{W}=\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-2}$
in $MKS$ system = in $CGS$ system
$ \mathrm{M}(\mathrm{kg})=10^{3} \mathrm{M}(\mathrm{gm}) $
$ \mathrm{L}(\mathrm{m}) =10^{2} \mathrm{~L}(\mathrm{~cm}) $
$\mathrm{T}(\mathrm{s}) =10^{0} \mathrm{~T}(\mathrm{~s}) $
$ \mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-2} =\left(10^{3} \mathrm{M}\right)^{1}(10^{2} \mathrm{~L})^{2}\left(10^{0} \mathrm{~T}\right)^{-2}$
$ =10^{3} \times 10^{4} \mathrm{M}^{1} \mathrm{~L}^{2}\mathrm{~T}^{-2}$
$ =10^{7} \mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-2}$
So, $MKS$ unit of work $=10^{7}$ CGS unit of work $\therefore 1 \mathrm{~J}=10^{7} \mathrm{erg}$
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- [IIT 2022]
Match List $I$ with List $II$ and select the correct answer using the codes given below the lists :
List $I$
List $II$
$P.$ Boltzmann constant
$1.$ $\left[ ML ^2 T ^{-1}\right]$
$Q.$ Coefficient of viscosity
$2.$ $\left[ ML ^{-1} T ^{-1}\right]$
$R.$ Planck constant
$3.$ $\left[ MLT ^{-3} K ^{-1}\right]$
$S.$ Thermal conductivity
$4.$ $\left[ ML ^2 T ^{-2} K ^{-1}\right]$
Codes: $ \quad \quad P \quad Q \quad R \quad S $
Match List $I$ with List $II$ and select the correct answer using the codes given below the lists :
| List $I$ | List $II$ |
| $P.$ Boltzmann constant | $1.$ $\left[ ML ^2 T ^{-1}\right]$ |
| $Q.$ Coefficient of viscosity | $2.$ $\left[ ML ^{-1} T ^{-1}\right]$ |
| $R.$ Planck constant | $3.$ $\left[ MLT ^{-3} K ^{-1}\right]$ |
| $S.$ Thermal conductivity | $4.$ $\left[ ML ^2 T ^{-2} K ^{-1}\right]$ |
Codes: $ \quad \quad P \quad Q \quad R \quad S $
- [IIT 2013]
The dimension of the ratio of magnetic flux and the resistance is equal to that of :
The dimension of the ratio of magnetic flux and the resistance is equal to that of :
If orbital velocity of planet is given by $v = {G^a}{M^b}{R^c}$, then
If orbital velocity of planet is given by $v = {G^a}{M^b}{R^c}$, then