Question

Water flows smoothly in a horizontal pipe. Figure 14-27 shows the kinetic energy Kof a water element as it moves along an xaxis that runs along the pipe. Rank the three lettered sections of the pipe according to the pipe radius, greatest first.

Short answer

The rank of the lettered sections of the pipe according to the pipe radius, greatest first is$B>C>A$ .

Step-by-step solution

The given data 

Rank of KE:$K{E}_A>K{E}_C>K{E}_B$

Understanding the concept of the kinetic energy

This problem is based on the continuity equation. Kinetic energy can be written in the form of the area by using a continuity equation. Thus, using this relation, we can get the required rank value.

Formulae:

The rate of flow of the fluid in terms of area and speed,$R_v=Av$ (i)

The kinetic energy of a body in motion,$KE=\left(\frac{1}{2}\right)m{v}^2$ (ii)

The mass of a body in terms of density,$m=\rho V$ (iii)

Calculation of the rank of the sections of the pipe according to the radius of the pipe

Substituting the value of mass from equation (iii) and velocity from equation (i) in equation (ii), we can get the kinetic energy equation as follows:

$$\begin{gathered} KE=\left(\frac{1}{2}\right)pV\times {\left(\frac{R_v}{A}\right)}^2 \\ =\left(\frac{\rho V{R}_v^2}{2{\mathrm{\pi }}^2}\right)\times \left(\frac{1}{r^4}\right)...............\left(b\right)\left(\because A={\mathrm{\pi r}}^2\right) \end{gathered}$$

This equation shows that KE is inversely proportional to$4^{th}$ power of the radius of the pipe. Thus, KE is minimum when radius will be maximum.

Therefore, rank of sections according to radius, greatest first is$B>C>A$ .