Question

If dimensions of critical velocity $v_c$ of a liquid flowing through a tube are expressed as $\left[{\eta }^x{\rho }^y{r}^z\right]$, where $\eta ,\rho$ and $r$ are the coefficient of viscosity of liquid, density of liquid and radius of the tube respectively, then values of $x,y$ and $z$ are given by

• A. 1, 1, 1
• B. 1, -1, -1
• C.
  -1, -1, 1
• D. -1, -1, -1

Answer 1

The correct option is B 1, -1, -1

$\left[v\right]=\left[L{T}^{-1}\right]$
$\eta =\frac{F}{A\frac{dv}{dx}}$
$\Rightarrow \left[\eta \right]=\frac{\left[ML{T}^{-2}\right]}{\left[L^2\right]\left[T^{-1}\right]}=\left[M^1{L}^{-1}{T}^{-1}\right]$
$\left[\rho \right]=\left[M{L}^{-3}\right]$
$\Rightarrow \left[L{T}^{-1}\right]=\left[M^1{L}^{-1}{T}^{-1}{]}^x\right[M{L}^{-3}{]}^y[L{]}^z$
Equating the exponents of $M,L$ and $T$ on both LHS and RHS
$\Rightarrow M^0=M^{(x+y)}$
$\Rightarrow y=-x$
For T,
$-1=-x$
x = 1
$y=-x=-1$
For L
$1=-x-3y+z$
$1=2+z\to z=-1$
$\Rightarrow z=-1$