Question

use dimensional analysis to check if the expression P=(yv^2)/2 + ygh is correct. Given that P is pressure in pascal, y is density in kgm^-3 , v is velocity in m/s, g is acceleration due to gravity in m/s^2 and h is the height in m(metre).​

Answer 1

Answer: The expression is correct

Explanation:

We have the following expression:

$P=\frac{yv^{2}}{2} + ygh$

Where:

Pressure: $P$ in units of Pascal ($Pa$)

$1 Pa= 1 \frac{N}{m} = \frac{kg m/s^{2}}{m}=\frac{kg}{ms^{2}}$

Density: $y$ in units of $\frac{kg}{m^{3}}$

Velocity: $v$ in units of $\frac{m}{s}$

Acceleration due to gravity: $g$ in units of $\frac{m}{s^{2}}$

Height: $h$ in units of $m$

Knowing this, let's begin with the dimensional analysis:

$Pa=\frac{(\frac{kg}{m^{3}}){(\frac{m}{s})}^{2}}{2} + (\frac{kg}{m^{3}})(\frac{m}{s^{2}})(m)$

$Pa=\frac{\frac{kg}{ms^{2}}}{2} + \frac{kg}{ms^{2}}$

$Pa=\frac{1}{2} \frac{kg}{ms^{2}} + \frac{kg}{ms^{2}}$

Remembering $1 Pa=\frac{kg}{ms^{2}}$, we are able to know this expression is correct.