Question

If dimensions of critical velocity $v_c$ of a liquid flowing through a tube are expressed as $\left[{\eta }^x{p}^y{r}^z\right]$, where $\eta ,p$ 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

Answer: (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

$1=-x-3y+z\to -1+3+z$

$1=2+z\to z=-1$

$\Rightarrow z=-1$