Problem Statement
The 1m diameter log (sp gr = 0.82) divides two shallow ponds as shown in the figure. Compute the net horizontal and vertical reactions at point C if the log is 3.7m.
Solution
1. Horizontal Force \( F_{H1} \) and \( F_{H2} \)
Horizontal force on ADC:
\( F_{H1} = \gamma \cdot A_1 \cdot \bar{y}_1 = 9810 \cdot 3.7 \cdot 1 \cdot 1.2 = 43556 \, \text{N (right)} \)
Horizontal force on BC:
\( F_{H2} = \gamma \cdot A_2 \cdot \bar{y}_2 = 9810 \cdot 3.7 \cdot 0.5 \cdot \frac{0.5}{2} = 4537 \, \text{N (left)} \)
2. Vertical Force \( F_{V1} \), \( F_{V2} \), and Weight of the Log \( W \)
Vertical force on ADC:
\( F_{V1} = \gamma \cdot \text{Volume}_{AOCD} = 9810 \cdot \frac{1}{2} \cdot \pi \cdot (0.5)^2 \cdot 3.7 = 14254 \, \text{N (up)} \)
Vertical force on BC:
\( F_{V2} = \gamma \cdot \text{Volume}_{BOC} = 9810 \cdot \frac{1}{4} \cdot \pi \cdot (0.5)^2 \cdot 3.7 = 7127 \, \text{N (up)} \)
Weight of the log:
\( W = \gamma_{\text{log}} \cdot \text{Volume}_{\text{log}} = 0.82 \cdot 9810 \cdot \pi \cdot (0.5)^2 \cdot 3.7 = 23376 \, \text{N (down)} \)
3. Reactions at Point C
Horizontal reaction at C:
\( R_x = F_{H1} – F_{H2} = 43556 – 4537 = 39019 \, \text{N (left)} \)
Vertical reaction at C:
\( R_y = W – F_{V1} – F_{V2} = 23376 – 14254 – 7127 = 1995 \, \text{N (up)} \)
Result:
- Horizontal Reaction: \( R_x = 39019 \, \text{N (left)} \)
- Vertical Reaction: \( R_y = 1995 \, \text{N (up)} \)
Explanation
- Horizontal Force: The horizontal forces result from the pressure of the water acting on the vertical projections of ADC and BC. These forces are calculated using the hydrostatic pressure formula and the respective areas and centroids.
- Vertical Force: The vertical forces are due to the weight of the water volumes above the curved surfaces ADC and BC. These volumes are computed based on the geometric properties of the log and the water distribution above it.
- Weight of the Log: The log’s weight is calculated using its specific gravity and volume, which counteracts the vertical forces from the water.
Physical Meaning
This problem demonstrates the interactions between hydrostatic pressure and the geometry of submerged objects:
- Horizontal Reaction: Represents the net force exerted horizontally on the log due to the water pressure differences on ADC and BC.
- Vertical Reaction: Balances the forces due to the water above the log and the log’s own weight, ensuring stability.
- Applications: Such calculations are critical in the design of structures like logs, barriers, or submerged pipelines, ensuring they can withstand the forces of water without losing stability.
