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Theoretical Determination of X Theory Shows That X X̄

This document presents equations and calculations for determining the center of pressure (Xp) of a submerged rectangular surface both theoretically and experimentally. Theoretically, Xp is calculated based on the density of the fluid, gravitational acceleration, depth to the centroid, and immersed area. Experimentally, Xp is determined by balancing torque measurements and geometrical relationships when the apparatus is in equilibrium. Sources of error between the theoretical and experimental values are believed to be inaccurate fluid height measurements and non-level positioning of the apparatus.

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91 views3 pages

Theoretical Determination of X Theory Shows That X X̄

This document presents equations and calculations for determining the center of pressure (Xp) of a submerged rectangular surface both theoretically and experimentally. Theoretically, Xp is calculated based on the density of the fluid, gravitational acceleration, depth to the centroid, and immersed area. Experimentally, Xp is determined by balancing torque measurements and geometrical relationships when the apparatus is in equilibrium. Sources of error between the theoretical and experimental values are believed to be inaccurate fluid height measurements and non-level positioning of the apparatus.

Uploaded by

salman
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as DOCX, PDF, TXT or read online on Scribd
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F = ρgћA

ρ = density of fluid
g = acceleration due to gravity
ћ = depth to centroid of immersed surface
A = area of immersed surface
Depth, d = 0.1m
Base, b = 0.075m
Distance from pivot to hanger, s = 0.25m
Distance from pivot to top of rectangular
surface, r = 0.1m

Theoretical determination of Xp
𝐼𝐶𝐺
Theory shows that Xp = x̄ =
𝐴x̄
ћ
where x̄ = distance from O to the centroid CG of the surface = 𝑐𝑜𝑠 θ

ɵ (angle degree) = 0 cos(0) = 1 x̄ = ћ


and ICG = 2nd moment of area of the immersed surface about the horizontal axis through CG

Experimental determination of Xp
for equilibrium of the experimental apparatus, moments about the pivot P give
F.Y = Mg.s
Where
y = distance from pivot to centre of pressure
M= mass added to hanger
S = distance from pivot to the hanger

Therefore
𝑀𝑔𝑠
Y= 𝐹
ℎ1
But Y = Xp + r - 𝑐𝑜𝑠θ
ℎ1
Therefore Xp = y – (r - )
𝑐𝑜𝑠θ

Where
r = distance from picot to top of rectangular surface
h1 = depth of water surface from top of rectangular surface
θ = angle of inclination of rectangular surface

x̄
ћ= θ=0 𝑐𝑜𝑠θ = 1 ћ = x̄
𝑐𝑜𝑠θ

𝑏𝑑³ 0.075 𝑥 0.1³


ICG = 12
= 12
= 6.25 x 10-6
In summing the moments about the pivot of the apparatus, the buoyant force is neglected. As seen
in theapparatus setup in Figure 2, the fluid resides inside the torus. The presence of buoyancy comes
from the air outsideof the torus. Because the density of air is a mere fraction of that of the material
of the torus and the fluid it contains,it can be neglected in the hydrostatic force calculations. The
weight of the torus can also be neglected. Because thecenter of the curvature of the torus is at the
location of the pivot, it is negated. The weight of the torus was notincluded in the calculations
because the device was calibrated with ballast water so as to begin the experiment witha net
moment of zero about the pivot. It was noted that a large discrepancy between the theoretical
andexperimental values occurred. This is most likely due to errors in measurement of the height of
the fluid inside of the torus. Another possible cause could be that the apparatus was not sitting level
on the counter where theexperiment was performed. If the apparatus is not sitting level, the
moment calculations will yield inaccurateresults. A leveling device on or near the testing apparatus
would aid in ensuring the moment balance is accurate.Another source of error would be the use of
the accepted density of water, 1000kg/m

, for the theoreticalcalculation of the hydrostatic force. This accepted value is the density of sea
water at 4

C. The water used in thisexperiment was tap water at approximately 23

C. However, if the actual density of the tap water was used, thetheoretical calculations would not
differ greatly enough to compensate for the magnitude of the error.

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