Question

A river flows down towards a lake along an incline. initially the river is 90 m above the lake flowing at a rate of 3 m/s. at the lake the velocity is 0 m/s. you are interested in the change of total energy per unit mass of the water. draw a schematic of the system, clearly identifying the system, and determine the change of total energy per unit mass of the water between the inlet and outlet of the system. assume that the internal energy of the water remains constant and that the pressure at the inlet and outlet is the same.

Answer 1

Total energy =kinetic energy +potential energy

Change in energy =change in (kinetic energy +potential energy)

potential energy, $P.E=mgh$

where m is the mass, g is acceleration due to gravity and h is the height.

potential energy per unit mass =gh

change in potential energy per unit mass = $\Delta P.E.=g\Delta h$

where, h is the height.

kinetic energy= $K.E. =\frac{1}{2}mv^2$

change in kinetic energy per unit mass,$\Delta K.E. =\frac{1}{2}\Delta v^2$

In the given question:

Height varies from 90 m to zero as river flows from 90 m height to lake at 0 m

Velocity varies from 3m/s at top to o m/s at bottom.

Therefore,

$\Delta E =\Delta K.E.+\Delta P.E.\\ \Rightarrow \Delta E/m=\frac{1}{2}\Delta v^2+g\Delta h=\frac{1}{2}(0^2-3^2) m^2s^{-2}+9.8 (0-9)m^2s^{-2}=(-\frac{9}{2}-88.2)m^2s^{-2}=-92.7 m^2s^{-2}$

Here, it was mentioned in the question internal energy of the water is constant and there is no change in the pressure at the inlet and outlet.