Solution Found!
CP CALC A cylindrical container of an incompressible
Chapter 12, Problem 86P(choose chapter or problem)
A cylindrical container of an incompressible liquid with density rotates with constant angular speed
about its axis of symmetry, which we take to be the y-axis (Fig. P12.86).
(a) Show that the pressure at a given height within the fluid increases in the radial direction (outward from the axis of rotation) according to \(\delta p / \delta r=\rho \omega^{2} r\)
(b) Integrate this partial differential equation to find the pressure as a function of distance from the axis of rotation along a horizontal line at y=0.
(c) Combine the result of part (b) with Eq. (12.5) to show that the surface of the rotating liquid has a parabolic shape; that is, the height of the liquid is given by \(h(r)=w^{2} r^{2} / 2 g\) (This technique is used for making parabolic telescope mirrors; liquid glass is rotated and allowed to solidify while rotating.)
Equation Transcription:
Text Transcription:
\delta p / \delta r=\rho \omega^2 r
y=0
h(r)=w^2 r^2 / 2 g
Questions & Answers
QUESTION:
A cylindrical container of an incompressible liquid with density rotates with constant angular speed
about its axis of symmetry, which we take to be the y-axis (Fig. P12.86).
(a) Show that the pressure at a given height within the fluid increases in the radial direction (outward from the axis of rotation) according to \(\delta p / \delta r=\rho \omega^{2} r\)
(b) Integrate this partial differential equation to find the pressure as a function of distance from the axis of rotation along a horizontal line at y=0.
(c) Combine the result of part (b) with Eq. (12.5) to show that the surface of the rotating liquid has a parabolic shape; that is, the height of the liquid is given by \(h(r)=w^{2} r^{2} / 2 g\) (This technique is used for making parabolic telescope mirrors; liquid glass is rotated and allowed to solidify while rotating.)
Equation Transcription:
Text Transcription:
\delta p / \delta r=\rho \omega^2 r
y=0
h(r)=w^2 r^2 / 2 g
ANSWER:Solution 86P
Step 1:
If the cylinder with an incompressible liquid is rotating through an axis, then, the liquid level in the cylinder will rise radially.
So, we can represent the change in pressure as,
But we know that,
Where, Density of the liquid
a - acceleration of the liquid
dr - elemental rise of liquid
Therefore we can write,
That is,
