The two dams are identical with the exception that the water reservoir behind dam A extends twice the horizontal distance behind it as that of dam B . Which one of the following statements regarding these two dams is correct ?

Options

A. The force exerted by the water on dam A is greater than that on dam B .
B. The force exerted by the water on dam B is greater than that on dam A .
C. Dam A is more likely to collapse than dam B if the water level rises .
D. The horizontal distance of the water behind the two dams does not determine the force on them .

Answer

To solve the problem regarding the two dams, we need to analyze the forces acting on each dam due to the water pressure. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Setup
We have two identical dams, dam A and dam B. The only difference is that the water reservoir behind dam A extends twice the horizontal distance compared to dam B.

### Step 2: Identify the Pressure on the Dams
The pressure exerted by a fluid at a certain depth is given by the formula:
\[ P = \rho g h \]
where:
- \( P \) is the pressure,
- \( \rho \) is the density of the fluid (water),
- \( g \) is the acceleration due to gravity,
- \( h \) is the height of the water column above the point in question.

### Step 3: Calculate Average Pressure
For both dams, assuming the height of the water column is the same (let's denote it as \( h \)), the average pressure at the base of each dam can be calculated. Since both dams have the same height of water:
- Average pressure on dam A: \( P_A = \frac{1}{2} \rho g h \)
- Average pressure on dam B: \( P_B = \frac{1}{2} \rho g h \)

### Step 4: Determine the Area of the Dams
The area of the dam wall that the water pressure acts upon is the same for both dams since they are identical in structure. Let’s denote the area as \( A \).

### Step 5: Calculate the Force on Each Dam
The force exerted by the water on each dam can be calculated using the formula:
\[ F = P \times A \]
Thus, for both dams:
- Force on dam A:
\[ F_A = P_A \times A = \left(\frac{1}{2} \rho g h\right) \times A \]
- Force on dam B:
\[ F_B = P_B \times A = \left(\frac{1}{2} \rho g h\right) \times A \]

### Step 6: Compare the Forces
From the calculations, we see that:
\[ F_A = F_B \]
This indicates that the force exerted by the water on both dams is the same, regardless of the horizontal distance of the water reservoir behind each dam.

### Conclusion
The correct statement regarding the two dams is that the horizontal distance of the water behind the two dams does not determine the force on the dam.

### Final Answer
The correct option is: The horizontal distance of the water behind the two dams does not determine the force on the dam.

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