H11
Centre of Pressure
Apparatus
© TQ Education and Training Ltd 2000
No part of this publication may be reproduced or transmitted in
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photocopy, recording or any information storage and retrieval
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Training Limited.
All due care has been taken to ensure that the contents of this
manual are accurate and up to date. However, if any errors >are
discovered please inform TQ so the problem may be rectified.
A Packing Contents List is supplied with the equipment. Carefully
check the contents of the package(s) against the list. If any items
are missing or damaged, contact your local TQ agent or TQ
immediately.
116
SECTION 1.0 INTRODUCTION
Figure 1 H11 Apparatus
The TQ H11 Centre of Pressure Apparatus permits the moment due to the fluid thrust on a wholly or partially
submerged plane surface to be measured directly and compared with theoretical analysis. Provision is made for
varying the inclination of the plane surface subjected to the fluid pressure so that the general case may be studied.
1.1 Description of Apparatus
Water is contained in a quadrant of a semi-circular Perspex tank assembly which is allowed to swing on a smooth bar.
The cylindrical sides of the quadrant have their axes coincident with the centre of rotation of the tank assembly, and
therefore the total fluid pressure acting on these surfaces exerts no moment about that centre. The only moment present
is that due to the fluid pressure acting on the plane surface. This moment is measured experimentally by applying weights
to a weight hanger mounted on the opposite side to the quadrant tank.
A second tank, situated on the same side of the assembly as the weight hanger, provides a trimming facility and enables
different angles of balance to be achieved.
The angular position of the plane and the height of water above it are measured on an angular scale engraved on the tank
and a linear scale on the back panel.
Included with the apparatus are base leveling feet and a spirit level.
1.2 Installation and Preparation
With the quadrant tank of the hopper fabrication held to the right-hand side, fit the hopper fabrication onto the centre
support of the main frame assembly.
Level the base plate by screwing the adjustable feet and observing the spirit level.
117
TO H11 Centre of Pressure Apparatus
A colored dye is supplied with the apparatus, which can be added to the water to help observation during
experimentation. It should only be necessary to add small amounts of dye (one or two capfuls) to produce suitable
coloring.
The apparatus should be cleaned regularly using soapy water. When cleaning, care should be taken to ensure that no
traces of grit or any hard abrasive materials are present on the cloth used. Do not use any materials containing man-made
fibers as these will scratch the Perspex.
118
SECTION 2.0 THEORY
2.1 Definition of Centre of Pressure
Centre of pressure may be defined as the point in a plane at which the total fluid thrust can be said to be acting normal
to that plane.
2.2 Analysis
1.
------------------ ~_~h
Figure 2 Schematic of Centre of Pressure Apparatus
The following analysis applies to the condition of a plane surface at various angles when it is wholly or partially
submerged in a fluid.
Let breadth of quadrant = B
and weight per unit volume = ω
Referring to Figure 2, consider an element at start depth y, width oy.
Force on element of = ω(y cosθ - h)Bdy
and the Moment of Force on element about 0 = ωB(y cosθ - h)yδy
Therefore, total moment about 0 = M = ω B ∫(cosθy2 - hy)dy
119
TO H11 Centre of Pressure Apparatus
Case 1: Plane Fully Submerged
Limits R1 and R2
𝑅2
𝑀 = 𝜔𝐵 ∫ (𝑐𝑜𝑠𝜃𝑦 2 − ℎ𝑦) 𝑑𝑦
𝑅1
𝑅
𝑀 = 𝜔𝐵 [
2
𝑐𝑜𝑠𝜃𝑦 3 ℎ𝑦 2
−
+ 𝑐]
3
2
𝑅
1
𝑀=
𝜔𝐵𝑐𝑜𝑠𝜃 3
𝜔𝐵 2
(𝑅2 − 𝑅13 ) −
(𝑅2 − 𝑅12 )ℎ
3
2
This equation is of the form of y = mx + c. A plot of M against h will yield a straight line graph of gradient
𝜔𝐵
− (𝑅22 − 𝑅12 ). The value of ω can now be calculated.
2
Case 2: Plane Partially Submerged
Limits R2 and h secθ
Hence:
R2
𝑀=∫
(𝑐𝑜𝑠𝜃𝑦 2 − ℎ𝑦)𝑑𝑦
ℎ𝑠𝑒𝑐𝜃
R
2
cosθy3 hy2
M = ωB [
−
+ c]
3
2
hsecθ
M=
ωBcosθR32 3
ωBh 2
(R2 − h3 sec3 θ) −
(R2 − h2 sec 2 θ)
3
2
M=
ωBcosθR32 ωBsec 2 θh3 ωBR22 h ωBsec 2 θh3
−
−
+
3
3
2
2
M=
ωBcosθR32 ωBR22 h ωBsec2 θh3
−
+
3
2
6
Rearranging:
M+
ωBR22 h ωBcosθR32 ωBsec2 θh3
=
+
2
3
6
ω can be obtained from Case I and h3 can then be plotted against M +
ωBR22 h
2
120
SECTION 3.0 EXPERIMENTAL PROCEDURE
Set up the equipment as previously described in Section 1.2, and affix the weight hanger to the hanger support located
on the top left of the hopper. The apparatus will now require trimming in order to bring the submerged plane to the
vertical (i.e. 0° position). This is achieved by gently pouring water into the trim tank until the desired position is achieved.
The horizontal line on the tank assembly should be read against 'the zero line on the back scale.
Add a 20 g weight to the weight hanger. Pour water, with dye added if necessary, into the quadrant tank until a 0° balance
is restored. Note the weight and the height reading of the water (h). Repeat the procedure for the full range of weights.
Empty both tanks of water. Again, with the weight hanger alone in position, trim the assembly by gently adding water
to the trim tank until a 30° balance is achieved. Add a 20 g weight, restore balance to 30° point and record values for h
for the full range of weights.
Readings may be tabulated in the form outlined in Table 1 and the results calculated in line with the theory given in
Section 2.
ω
(gm)
M
𝑊𝑥9.81𝑥𝑅3
103
(𝑁𝑚)
h
(mm)
h
(m)
θ°
h3
3
(m )
𝑀+
𝜔𝐵𝑅22 ℎ
2
(Nm)
Table 1 Format of results
121
TO H11 Centre of Pressure Apparatus
Figures 3 and 4 show the general form of the graphs expected from this experiment.
h(m)
Figure 3 Graph showing plane fully submerged
Figure 4 Graph showing plane partially submerged
The results obtained from this experiment tend to agree very well with results calculated from hydrostatic theory.
Disagreement between observation and theory is likely to be due to error in measurement or the construction tolerances
of the apparatus rather than to any imprecision of the theory.
122
Product Specification
PRODUCT
CODE NUMBER
DENSITY g/ml @ 20·C
SOLVENT
PRESERVATIVE
COLOURING
APPEARANCE
STORAGE/SHELF LIFE
Bri11iant B1ue FCF Co10ur Liquid 1% Soluti0n
BZ2290
1. 003 ± o. 005
Water
1000 ± 100 mg/liter benzoic acid
Contains: Brilliant Blue FCF (133)
The product -is an intensely deep blue colored
liquid. The color of an 0.1% v/v aqueous
solution will compare favorably in shade and
intensity of color with previous deliveries or
samples.
Stored in cool conditions and away from light in
the original, full, tightly sealed container,
this product has a shelf life of at least 12
months.
LEGISLATION
DATE OF ISSUE
COMPILATION NO.
This product conforms in all respects with the
current UK Food and Drinks Legislation. Further
legislative information is available on request.
18th July, 1995
1
123
Health & Safety Data Sheet
Product:
Referance:
Brilliant Blue Color Solution
BZ 2290
Storage:
Care should be taken ill storing and handling the material. It is good
general practice to store in full, tightly sealed containers away from
sources of heat and extremes of temperature in a strictly no smoking area.
Containers should never be stored ill direct sunlight.
This product will stain skin and clothing.
Avoid contact with skin and eyes by wearing suitable protective clothing, PVC
gloves and eye protection. Normal standards of hygiene and regular washing
with water should also prevent irritation.
Do not ingest the undiluted color only use at the recommended dose rate. If
accidentally swallowed rinse mouth with water and seek medical advice. If
accidentally splashed into eyes irrigate with plenty of water for at least
ten minutes.
The handling area should be well ventilated and sound hygiene practice should
be adopted at all times in keeping with the handling of any foodstuff
ingredient.
Spillage
Contain the spillage with an inert material such as sand or earth and remove
into a suitable container. Wash the-· contaminated area with detergent and
water. Any waste material should be disposed of in compliance with the UK
Control of Pollution Act 1974.
18th July, 1995
124