Abstract

 Abstract: The main objective of this experiment is to determine the center of  pressure on a plane plane surface (rectangular (rectangular surface) of the torroid torroid by comparing the normal force (exerted by the liquid) and the weights on the balance bar.

 Introduction: When a liquid is in contact with a solid surface, then two forces will be formed: 1. Shear force: which is caused due to the viscosity of the liquid, and it describes the resistance of ow (friction) and depends on the type of the liquid.  ! " (du#dy) Where:  : Shear stress ($a).  ": %iscosity ($a.s).  du#dy : %elocity &radient (1#s).

'. ormal force: it is caused from the wei&ht of liquid on a plane around the surface, as dams and it acts on the

surface as a line (normal to the area) and the centroid area of the surface. Sometimes we call the normal force, the pressure force.  ! $* and in the case of a static uid  ! +& hc *. Where hc: centroid.  his relation is valid if the shape is rectan&ular and the plane surface is hori-ontal.  or other shapes (rather than hori-ontal or rectan&ular) to nd the centroid, evaluate the inte&ral: /d ! /$d*. Where the pressure is linearly distributed over the surface. 0n this eperiment the non2hori-ontal surface is also a3ected by the hydrostatic force due to the static liquid (water).

 Apparatus and Procedures :  (a) Locate the torroid on the dowel pins and fasten to the balance arm by the central screw.  (b) Measure the dimensions a, b, and d, and the distance L from the knife – edge axis to the balance pan axis.  (c) Position the perspex tank on work surface and locate the balance arm on the knife edges.

 (d) ttach a length of hose to the drain cock and direct the other end of  hose to the sink. ttach a length of hose to tap !" and place the free end  in the triangular aperture on the top of the perspex tank. Le#el the tank, using the ad$ustable feet in con$unction with the spirit le#el. (e) d$ust the counter % balance weigh until the balance arm is hori&ontal. 'his is indicated on a gate ad$acent to the balance arm. (f)ill water to the perspex tank until the water is le#el with the bottom edge of the torroid.  (g) Place a mass on the balance pan and fill water to the tank until the balance arm is hori&ontal. ote the water le#el on the scale. ine ad$ustment of the water le#el may be achie#ed by o#er – filling and   slowly draining, using the drain cock.  (h) *epeat the procedure under section.  (g) for different masses +  masses for water le#els y - d (complete immersion) and  masses for y  d (partial immersion) (i) *epeat readings for reducing masses on the balance pan.

 Results and Discussion: For the apparatus used, the formula  F = PA= ρgya… … ( 1 )

!nd  y − y =

I   … … ( 2 )  Ay

"ay be applied to give expressions for the moment of the hydrostatic force about the #nife$edge axis, where

 : the water force of the torroid area.  y: the centroid of the area.  $: the water pressure at the centroid of the area.  y: center of pressure.  0: the second moment of the area.

* For Partial Immersion: y
 y − y =by / 12 / by / 2= y / 6 … . ( 4 )

 he moment 4 of  about 5nife2ed&e ais is &iven by:  M =½ ρgby ( a + d − y / 2 + y / 6 )=½ ρgby ( a + d − y / 3 ) …. ( 5 )

!lso  M = gml…. ( 6 )

%here

4: the mass added to the balance pan.  6: the distance from the 7nife2ed&e ais to balance pan suspension rod ais.

ml =½ ρby ( a +d − y / 3 ) … . ( 7 )

* For Complete Immersion: y>d (see fig.3)  F = ρgybd … . ( 8 )

%here  y = y −d / 2 …. ( 9 )

and  y – y =bd / 12 / bdy = d / 12  y … . ( 10 )

The moment " of F about #nife$edge axis for this case is given by  M = ρgybd ( a + d /2 + d /12  y ) … . ( 11)

ml = ρybd ( a + d /2 + d /12 y ) … . ( 12 )

&ncreasing Mass (kg) '.' '.' '.-

Y(m) '.' '.'+* '.'

M/y (kg/m) .*+ .- -.

Y(m)

M/y (kg/m)

8ecreasin&: Mass (kg)

9.9 9.9; 9.9<

9.91 9.9'1 9.9=;

 '.<; '.'

Discussion:

&t can be noticed that as the applied mass increases, the distances y and yc increase because the moment is needed for balance because the hydrostatic force exerted on the torroid increases.

*fter ndin& the slope and intercept for both partial and complete immersion there were some deviations between eperimental and theoretical values due to errors in measurements.

Conclusions and Recommendations: / &n the case of a static fluid, the pressure force for the hori0ontal face can be calculated from the relationship  F = ρg hc A . 1ut for a non$hori0ontal shape the centeroid is found from the integration and then the pressure force (hydrostatic force) is evaluated. / The location at which the resultant pressure force acts gives a moment  balance static field and the location is located under the centeroid  because the force increases with depth. / There are linearly distributions for the force over the surface (from the equations).

/ 2rrors 3n predictable fluctuations in the measured quantities,  personal errors, and inaccurate reading or scaling.

 References: 1)*bu2>adayil, ?.6aboratory 4anual for luid 4echanics @hA=BB, >CS, '99', 0rbid2>ordan. ') 0. Dobbert,Eluid 4echanics with An&ineerin& *pplicationsE=rd edition, 4cFraw Gill, 1HHB.