Problem 3.48
(Difficulty: 2)
3.48 Calculate the minimum force 𝑃 necessary to hold a uniform 12 𝑓𝑓 𝑠𝑠𝑠𝑠𝑠𝑠 gate weighing
500 𝑙𝑙𝑙closed on a tank of water under a pressure of 10 𝑝𝑝𝑝. Draw a free body of the gate as part of
your solution.
Given: All the parameters are shown in the figure.
Find: The minimum force 𝑃 to hold the system.
Assumptions: Fluid is static and incompressible
Solution: Apply the hydrostatic relations for pressure, force, and moments, with y measured from the
surface of the liquid:
𝑑𝑑
= 𝜌𝑔=𝛾
𝑑𝑑
𝐹𝑅 = � 𝑝 𝑑𝑑
A free body diagram of the gate is
𝑦 ′ 𝐹𝑅 = � 𝑦 𝑝 𝑑𝑑
The gage pressure of the air in the tank is:
𝑝𝑎𝑎𝑎 = 10 𝑝𝑝𝑝 = 1440
This produces a uniform force on the gate of
𝐹1 = 𝑝𝑎𝑎𝑎 𝐴 = 1440
𝑙𝑙𝑙
𝑓𝑓 2
𝑙𝑙𝑙
× (12 𝑓𝑓 × 12 𝑓𝑓) = 207360 𝑙𝑙𝑙
𝑓𝑓 2
This pressure acts at the centroid of the area, which is the center of the gate. In addition, there is a force
on the gate applied by water. This force is due to the pressure at the centroid of the area. The depth of
the centroid is:
𝑦𝑐 =
The force is them
𝐹2 = 𝛾ℎ𝑐 𝐴 = 62.4
12 𝑓𝑓
× sin 45°
2
𝑙𝑙𝑙 12 𝑓𝑓
×
× sin 45° × 12 𝑓𝑓 × 12 𝑓𝑓 = 38123 𝑙𝑙𝑙
𝑓𝑓 3
2
The force F2 acts two-thirds of the way down from the hinge, or 𝑦 ′ = 8 𝑓𝑓.
Take the moments about the hinge:
𝐿
𝐿
−𝐹𝐵 sin 45° + 𝐹1 + 𝐹2 × 8 𝑓𝑓 − 𝑃 × 12 𝑓𝑓 = 0
2
2
Thus
𝑃=
−500 𝑙𝑙𝑙 × 6 𝑓𝑓 × sin 45° + 207360 𝑙𝑙𝑙 × 6 𝑓𝑓 + 38123 𝑙𝑙𝑙 × 8 𝑓𝑓
= 128900 𝑙𝑙𝑙
12 𝑓𝑓