The vertical surface of a reservoir dam that is in contact with the water is 120 m wide and 12 m high . The air pressure is one atmosphere . Find the magnitude of the total force acting on this surface in a completely filled reservoir . (The pressure varies linearly with depth , so , you must use an average pressure)

Options

A. `6.8 xx 10^(8) N`
B. `3.1 xx 10^(8) N`
C. `4.6 xx 10^(8) N `
D. `2.3 xx 10^(8) N`

Answer

To find the total force acting on the vertical surface of a reservoir dam, we can follow these steps:

### Step 1: Understand the Problem
The vertical surface of the dam is 120 m wide and 12 m high. The pressure exerted by the water on the dam increases linearly with depth due to the weight of the water above. We need to calculate the total force acting on this surface.

### Step 2: Determine the Pressure at Depth
The pressure at a depth \( x \) in a fluid is given by the formula:
\[ P = \rho g x \]
where:
- \( \rho \) is the density of the water (approximately \( 1000 \, \text{kg/m}^3 \)),
- \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)),
- \( x \) is the depth below the surface of the water.

### Step 3: Calculate the Average Pressure
Since the pressure varies linearly, the average pressure \( P_{avg} \) over the height of the dam can be calculated as:
\[ P_{avg} = \frac{P_{top} + P_{bottom}}{2} \]
At the top (depth \( x = 0 \)):
\[ P_{top} = \rho g (0) = 0 \, \text{Pa} \]
At the bottom (depth \( x = 12 \)):
\[ P_{bottom} = \rho g (12) = 1000 \times 9.81 \times 12 \]
Calculating \( P_{bottom} \):
\[ P_{bottom} = 1000 \times 9.81 \times 12 = 117720 \, \text{Pa} \]
Thus, the average pressure is:
\[ P_{avg} = \frac{0 + 117720}{2} = 58860 \, \text{Pa} \]

### Step 4: Calculate the Area of the Dam Surface
The area \( A \) of the vertical surface of the dam is given by:
\[ A = \text{width} \times \text{height} = 120 \, \text{m} \times 12 \, \text{m} = 1440 \, \text{m}^2 \]

### Step 5: Calculate the Total Force
The total force \( F \) acting on the surface can be calculated using the formula:
\[ F = P_{avg} \times A \]
Substituting the values:
\[ F = 58860 \, \text{Pa} \times 1440 \, \text{m}^2 \]
Calculating \( F \):
\[ F = 58860 \times 1440 = 84614400 \, \text{N} \]

### Final Answer
The magnitude of the total force acting on the vertical surface of the dam is:
\[ F \approx 84.6 \, \text{MN} \, (\text{MegaNewtons}) \]

---