Find the primary dimensions of each of the following terms. a. (pV2)/2 (kinetic pressure), where p is fluid density and V is velocity b. T (torque) c. P (power) d. (PV2L)/ (Weber number), where p is fluid density, V is velocity, L is length, and is surface tension.

Solution

Verified$\textbf{a)}$

In order to determine the primary dimension of the term under the section $\textbf{a)}$ we will consider the base units for every variable. Since the unit used for the density is:

$\mathrm{\frac{kg}{m^3}}$

Therefore, we can write the primary dimension for the density, using the relation between the signs and the dimensions of given base units as:

$[\mathrm{\frac{kg}{m^3}}]-\mathrm{\frac{mass}{volume}}-\mathrm{\frac{M}{L^3}}$

The unit used for the velocity is:

$\mathrm{\frac{m}{s}}$

We are going to write the primary dimension for the velocity, using the relation between the signs and the dimensions of given base units as:

$\mathrm{\frac{m}{s}}-\mathrm{\frac{length}{time}}-\mathrm{\frac{L}{T}}$

Finally we are able to determine the primary dimension of the kinetic pressure as:

$\begin{align*} \frac{\rho\cdot{V^2}}{2}&-\mathrm{\frac{M}{L^3}\cdot\left(\frac{L}{T}\right)^2}\\ &-\mathrm{\frac{M}{L^3}\cdot\frac{L^2}{T^2}}\\ &-\mathrm{\frac{M}{L\cdot{T^2}}} \end{align*}$

$\boxed{\mathrm{\frac{M}{L\cdot{T^2}}}}$