Problem Statement
Find the net horizontal and vertical forces acting on the surface ABCDEF of width 5 m as shown in the figure below. BCD is a half-circle.
Solution
1. Forces on Inclined Surface AB
Length of AB:
\( AB = \frac{2}{\sin 45^\circ} = 2.8284 \, \text{m} \)
Pressure force on AB:
\( F_1 = \gamma_{\text{water}} \cdot A_1 \cdot \bar{y}_1 \)
- \( \gamma_{\text{water}} = 9810 \, \text{N/m}^3 \)
- \( A_1 = 2.8284 \cdot 5 = 14.142 \, \text{m}^2 \)
- \( \bar{y}_1 = 1 \, \text{m} \)
Substituting:
\( F_1 = 9810 \cdot 14.142 \cdot 1 = 138733 \, \text{N} \)
Horizontal and vertical components:
- \( F_{1x} = F_1 \cos 45^\circ = 98099 \, \text{N (right)} \)
- \( F_{1y} = F_1 \sin 45^\circ = 98099 \, \text{N (up)} \)
2. Forces on Curved Surface BCD
- \( F_{2x} = \gamma_{\text{water}} \cdot A_2 \cdot \bar{y}_2 \)
- \( = 9810 \cdot (2 \cdot 5) \cdot 3 = 294300 \, \text{N (right)} \)
- \( F_{2y} = \gamma_{\text{water}} \cdot V_{\text{above BCD}} \)
- \( = 9810 \cdot \left( \frac{1}{2} \pi \cdot \frac{2^2}{4} \cdot 5 \right) = 77048 \, \text{N (down)} \)
3. Forces on EF
- Due to water: \( F_3 = 9810 \cdot (2.8284 \cdot 5) \cdot 5 = 693665 \, \text{N} \)
- \( F_{3x} = 490495 \, \text{N (right)} \)
- \( F_{3y} = 490495 \, \text{N (up)} \)
- Due to oil: \( F_4 = 0.8 \cdot 9810 \cdot (2.8284 \cdot 5) \cdot 1 = 110986 \, \text{N} \)
- \( F_{4x} = 78479 \, \text{N (left)} \)
- \( F_{4y} = 78479 \, \text{N (down)} \)
Net Forces:
- Net Horizontal Force: \( F_{\text{net, horizontal}} = 804415 \, \text{N (right)} \)
- Net Vertical Force: \( F_{\text{net, vertical}} = 433067 \, \text{N (up)} \)
Explanation
- Horizontal Forces: The horizontal forces are due to the water and oil pressure acting on the vertical and curved surfaces. The components of these forces are summed to calculate the net horizontal force.
- Vertical Forces: The vertical forces are due to the weight of the water above curved surfaces and the pressure acting on inclined surfaces. These are balanced to calculate the net vertical force.
Physical Meaning
- Structural Design: These calculations ensure the structural integrity of submerged or inclined surfaces, accounting for forces due to water and oil pressure.
- Practical Applications: Such problems are crucial in designing dams, tanks, and pipelines to resist pressure forces and ensure stability.
