Answer:
Let
\[\text{R}=\text{K}{{\upsilon
}^{\text{a}}}{{\rho }^{\text{b}}}{{\eta }^{\text{c}}}{{\text{D}}^{\text{1}}}\]
...(i)
where K is a dimensionless constant of proportionality and a, b, c
and 1 are the respective powers of
\[\upsilon\]
,
\[\rho\]
,
\[\eta\]
and D to define R
Writing the dimensions of various quantities in (16), we get
\[{{\text{M}}^{0}}{{\text{L}}^{0}}{{\text{T}}^{0}}={{\left[
\text{L}{{\text{T}}^{-\text{1}}} \right]}^{\text{a}}}{{\left[
\text{M}{{\text{L}}^{-\text{3}}} \right]}^{\text{b}}}{{\left[ \text{M}{{\text{L}}^{-\text{1}}}{{\text{T}}^{-\text{1}}}
\right]}^{\text{c}}}{{\text{L}}^{\text{1}}}={{\text{M}}^{\text{b}+\text{c}}}{{\text{L}}^{\text{a}-\text{3b}-\text{c}+\text{1}}}{{\text{T}}^{-\text{a}-\text{c}}}\]
Applying the principle of homogeneity of dimensions, we get
\[\text{b}+\text{c}=0\]
.....(ii)
\[\text{a}-\text{3b}-\text{c}+\text{1}=0\]
...(iii)
\[-\text{
a}-\text{c}=0\]
...(iv)
From (ii),
\[\text{b}=-\text{c}\]
,
From (iv),
\[\text{a}=-\text{c}\]
.
Put in (iii), \[-\text{c}\text{3}\left(
-\text{c} \right)-\text{c}+\text{1}=0\]
\[\text{c}+\text{1}=0\]
or
\[\text{c}=-\text{1}\]
\[\therefore\]
\[\text{a}=-\text{c}=+\text{1},\text{b}=-\text{c}=+\text{1}\]
Putting these values in (i), we get
\[\text{R}=\text{K }{{\upsilon
}^{\text{1}}}{{\rho }^{\text{1}}}{{\eta }^{-\text{1}}}{{\text{D}}^{\text{1}}}\]
\[\text{R}=\text{K}\frac{v\rho
D}{\eta }\text{ }\]
This is the required relation.
Note that the above question, R depends on four factors having
dimensions. The relation for R cannot be derived ordinarily. The dependence of
R on any one of the factors must be known/given, as in this question\[R\propto
{{D}^{1}}\] is given.
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