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Question: The dimensional formula of viscosity is \([M^1L^{-1}T^{-1}]\). One poise, i.e. C.G.S unit of viscosi...

The dimensional formula of viscosity is [M1L−1T−1][M^1L^{-1}T^{-1}]. One poise, i.e. C.G.S unit of viscosity is:
\eqalign{ & {\text{A}}.{\text{ }}{10^{ - 1}}M.K.S. \cr & {\text{B}}.{\text{ }}{10^0}M.K.S. \cr & {\text{C}}.{\text{ }}{10^1}M.K.S. \cr & {\text{D}}.{\text{ }}{10^{ - 2}}M.K.S. \cr}

Explanation

Solution

C.G.S units are the ones in which cm, gram and seconds are taken as base units. Whereas M.K.S units are those units in which kilogram, meter and second are taken as base units. To convert any quantity from one system of units to another system, we need to assume one unit as constant times other and then equating will give the answer.

Formula Used : 1kg = 1000g, 1m=100cm.

Complete step-by-step answer :
Let’s suppose, Viscosity in M.K.S is KK times viscosity in C.G.S units.
Hence we need to equate them.
[M1L−1T−1][M^1L^{-1}T^{-1}] = KK [Mc1Lc−1Tc−1][M_c^1L_c^{-1}T_c^{-1}] [where ‘c’ means cgs unit; mass, length and time in cgs unit]
Now putting values of different terms both with magnitudes and units:
[kg1m−1s−1]=K[g1cm−1s−1][ kg^1 m^{-1} s^{-1} ] = K [ g^1 cm^{-1} s^{-1} ]
Since we need to convert the data in C.G.S. units, therefore converting various units to C.G.S units:
[(1000g)1(100cm)−1s−1](1000g)^1 (100cm)^{-1} s^{-1}] = K[g1cm−1s−1]K [ g^1 cm^{-1} s^{-1} ]
Or [10001×100−1]g1cm−1s−1=K×g1cm−1s−11000^1 \times 100^{-1}] g^1 cm^{-1} s^{-1} = K \times g^1 cm^{-1} s^{-1}
= 1000100×g1cm−1s−1=K×g1cm−1s−1\dfrac{1000}{100} \times g^1 cm^{-1} s^{-1} = K \times g^1 cm^{-1} s^{-1}
= 10 ×g1cm−1s−1=K×g1cm−1s−1\times g^1 cm^{-1} s^{-1} = K \times g^1 cm^{-1} s^{-1}
Since both sides are the units are same, hence the values must also be equal.
Hence K=10.
Now as per our assumption, Viscosity in M.K.S is KK times viscosity in C.G.S units.
This means Viscosity in M.K.S is 10 times viscosity in C.G.S units.
Also given that viscosity in C.G.S is called poise, hence Viscosity in M.K.S is 10 poise.
Hence 1 poise is 110\dfrac1{10} M.K.S units or 10−110^{-1} M.K.S. units.
Option A. is correct.

Note :
The K is called the conversion factor which is used to convert the quantity from one unit to another. One can convert any unit to any system of units, by following the above process. One should remember the dimensional formulas for different quantities to proceed further. An important example of conversion factor is conversion of speed from kms−1km{s^{ - 1}} to ms−1m{s^{ - 1}}:1 kms−1=518ms−11{\text{ }}km{s^{ - 1}} = \dfrac{5}{{18}}m{s^{ - 1}}. For the conversion of any and all physical quantities in one unit system to another, one needs to just multiply them to the factor values of their derived quantities in the respective system they are being converted to.