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Explanation & Answer
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Question 1.
A vertical surface with height 2.5 feet and width 2 feet is submerged 1 foot below the water
surface. Find the resultant hydrostatic force and the location of the center of pressure.
𝜌𝑔ℎ1
Solution.
The pressure prism for the submerged vertical surface
is show in the diagram to the right.
ℎ = 2.5 𝑓𝑡
𝜌 = 1000𝑘𝑔/𝑚3
𝑔 = 9.81 𝑚/𝑠 2
ℎ1 = 1 𝑓𝑡
ℎ2 = ℎ + ℎ1 = 2.5 + 1 = 3.5 𝑓𝑡
𝑤𝑖𝑑𝑡ℎ = 2 𝑓𝑡
𝐴𝑟𝑒𝑎, 𝐴 = ℎ × 𝑤𝑖𝑑𝑡ℎ = 2.5 × 2 = 5 𝑓𝑡 2
𝐹1
ℎ
𝐹𝑅
𝐹2
The resultant force 𝐹𝑅 is given by the volume of the pressure
prism:
𝜌𝑔(ℎ2 − ℎ1 )
𝜌𝑔ℎ1
𝐹𝑅 = 𝐹1 + 𝐹2
𝐹1 = 𝜌𝑔ℎ1 ∙ 𝐴 = 1000 × 9.81 × 1 × 5 = 49,050
𝐹2 =
1
1
𝜌𝑔(ℎ2 − ℎ1 ) ∙ 𝐴 = × 1000 × 9.81 × (3.5 − 1) = 12,262.5
2
2
∴ 𝐹𝑅 = 49,050 + 12,262.5 = 61,312.5𝑁
The resultant force 𝑭𝑹 on the prism is 𝟔𝟏, 𝟑𝟏𝟐. 𝟓 𝑵 = 𝟔𝟏. 𝟑𝟏 𝒌𝑵
The location of the center of pressure 𝑦𝑐 from the top of the vertical surfacce is gotten from the
equation.
𝐹𝑅 ∙ 𝑦𝑐 = 𝐹1 ∙ 𝑦1 + 𝐹2 ∙ 𝑦2
𝑦1 =
ℎ 2.5
=
= 1.25,
2
2
𝑦2 =
2
2
ℎ = × 2.5 = 1.667
3
3
∴ (61,312.5)𝑦𝑐 = (49,050 × 1.25 + 12262.5 × 1.667)
61,312.5 𝑦𝑐 = 81,754.09
𝑦𝑐 =
81,754.09
= 𝟏. 𝟑𝟑𝟑 𝒇𝒕
61,312.5
𝐟𝐫𝐨𝐦 𝐭𝐡𝐞 𝐭𝐨𝐩 𝐨𝐟 𝐭𝐡𝐞 𝐯𝐞𝐫𝐭𝐢𝐜𝐚𝐥 𝐬𝐮𝐫𝐟𝐚𝐜𝐞.
From the water surface the location of pressure center, 𝒚𝒄 = 𝟏 + 𝟏. 𝟑𝟑𝟑 = 𝟐. 𝟑𝟑 𝒇𝒕 below
the water surface.
Question 2
A 10.0 m long vertical wall separates seawater (specific gravity S.G. =1.025) from fresh water. If
the depth of seawater is 8.0 m, what depth of fresh water is required to give zero resultant force?
Calculate the moments at the toe of the wall.
Solution
The diagram to the right shows the illustration of the set up.
For the resultant force 𝐹𝑅 to be zero, the force due to the
seawater 𝐹𝑤 must balance the force due to fresh water, 𝐹𝑤 .
𝐹𝑠 = 𝐹𝑤
Freshwater
SG = 1.000
8𝑚
𝜌𝑠 𝑔ℎ𝑠 ∙ 𝐴𝑠 = 𝜌𝑤 𝑔ℎ𝑤 ∙ 𝐴𝑤
Seawater
SG = 1.025
𝑥𝑚
𝜌𝑠 ℎ𝑠 ∙ 𝐴𝑠 = 𝜌𝑤 ℎ𝑤 ∙ 𝐴𝑤
10𝑚
𝜌𝑠 = 1.025 × 1000 = 1025 𝑘𝑔/𝑚3
ℎ𝑠 = 8
𝐴𝑠 = 8 × 10 = 80
𝜌𝑤 = 1000 𝑘𝑔/𝑚3
ℎ𝑤 = 𝑥
𝜌𝑠 𝑔ℎ1
𝜌𝑤 𝑔ℎ2
𝐴𝑠 = 𝑥 × ...