Relative Equilibrium of Liquids Relative equilibrium of liquid is a condition where the whole mass of liquid including the vessel in which the liquid is contained, is moving at uniform accelerated motion with respect to the earth, but every particle of liquid have no relative motion between each other. There are two cases of relative equilibrium that will be discussed in this section: linear translation and rotation. Note that if a mass of liquid is moving with constant speed, the conditions are the same as static liquid in the previous sections.
Formulas
or details of the following formulas see the translation translation and and rotation rotation pages. pages.
!ori"ontal #otion
$nclined #otion
%ertical #otion
Rotation and
Rectilinear Translation & #oving %essel Horizontal Motion
$f a mass of fluid moves hori"ontally along a straight line at constant acceleration acceleration a, the liquid surface assume an angle θ with the hori"ontal, see figure below.
or any value of a, the angle θ can be found by considering a fluid particle of mass m on the surface. The forces acting on the particle are the weight W = mg, inertia force or reverse effective force REF = ma, and the normal force N which is the perpendicular reaction at the surface. These three forces are in equilibrium with their force polygon shown to the right.
rom the force triangle
Inclined Motion
'onsider a mass of fluid being accelerated up an incline α from hori"ontal. The hori"ontal and vertical components of inertia force REF would be respectively, x = mah and y = mav.
rom the force triangle above
but a cos α = ah and a sin α = a v, hence
(se )*+ sign for upward motion and )+ sign for downward motion.
Vertical Motion
The figure shown to the right is a mass of liquid moving vertically upward with a constant acceleration a. The forces acting to a liquid column of depth h from the surface are weight of the liquid W = γV, the inertia force REF = ma, and the pressure F = pA at the bottom of the column.
(se )*+ sign for upward motion and )+ sign for downward motion. -lso note that a is positive for acceleration and negative for deceleration.
Rotation & Rotating %essel hen at rest, the surface of mass of liquid is hori"ontal at /0 as the figure. hen this mass of liquid is rotated about a vertical a1is at constant angular velocity ω radian per second, it will assume the surface -2' which is parabolic. Every particle is sub3ected to centripetal force or centrifugal force CF = mω2x which produces centripetal acceleration towards
the center of rotation. 4ther forces that acts are gravity force W = mg and normal force N.
shown in
here tan θ is the slope at the surface of paraboloid at any distance 1 from the a1is of rotation.
rom 'alculus, y5 6 slope, thus
or cylindrical vessel of radius r revolved about its vertical a1is, the height h of paraboloid is
Other Formulas
2y squaredproperty of parabola, the relationship of y, 1, h and r is defined by
%olume of paraboloid of revolution
$mportant conversion factor
Tags:
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centri!ga" orce
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ang!"ar ve"ocity
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para#o"oi$
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s%!are$ property o para#o"a
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centripeta" acce"eration
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s"ope o para#o"oi$
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vo"!me o para#o"oi$
Problem 1
- closed cylindrical vessel 7 m. in diameter and 8 m high is filled with water to a height of 9. m. The rest is filled with air, the pressure of which is ;< =/a. $f the vessel is rotated at ;>; rpm about its a1is, determine the ma1imum and minimum inside pressure at the base.
Solution 1
?peed of rotation
T$/: #ultiply rpm by & @7< for fast conversion to rad@sec. Notice that the above procedure is actually a multiplication of this amount.
hen 1 6 r 6 ;. m, y 6 h
Aetermine the position of the vorte1: )Note: The height of paraboloid is equal to !B @BA when the vorte1 touches the bottom of the tan=.+
?ince h C !B @BA, the vorte1 is below the vessel. ?ee figure below.
-t 1 6 1;, y 6 y;
-t 1 6 1B, y 6 y; * 8
%olume of air
inal volume of air 6 $nitial volume of air
The minimum pressure at the base occurs at all points within the circle of radius 1 ; and is equal to the original air pressure. answer
The ma1imum pressure will occur anywhere along the circumference of the base.
answer
