The vertical surface of a reservoir dam that is in contact with the water is 200 m wide and 14 m high. The air pressure is one atmosphere. Find the magnitude of the total force acting on this surface in a completely filled reservoir. (Hint: The pressure varies linearly with depth, so you must use an average pressure.)

average pressure: air pressure doesn't count, because it is on both sides of the dam.

avg pressure due to water: avg depth 14 m.

Water of 7 m high generates a pressure of
7m*densitywater*g=7*1010kg*9.8m/s^2* 1/area

pressure= you do it from above.

Total force= pressure*200*14 Netwons

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To find the magnitude of the total force acting on the vertical surface of the reservoir dam, we need to calculate the average pressure and then multiply it by the surface area.

1. Determine the average pressure:
The pressure varies linearly with depth, so the average pressure can be found by taking the average of the pressure at the top and bottom of the vertical surface.

At the top of the surface, the pressure is equal to the atmospheric pressure, which is one atmosphere.
At the bottom of the surface, the pressure is given by the hydrostatic pressure formula: P = ρgh, where
- ρ is the density of water (assumed to be 1000 kg/m^3 for freshwater)
- g is the acceleration due to gravity (assumed to be 9.8 m/s^2)
- h is the height of the vertical surface, which is 14 m.

Plugging in the values, we get:
P_bottom = (1000 kg/m^3)(9.8 m/s^2)(14 m) = 137,200 Pa

The average pressure is then:
P_average = (1 atm + 137,200 Pa) / 2 = 68,600 Pa

2. Calculate the surface area:
The surface area of the dam is given as 200 m wide and 14 m high.
A = width * height = 200 m * 14 m = 2800 m^2

3. Find the magnitude of the total force:
The force is equal to the pressure multiplied by the surface area.
F = P_average * A = 68,600 Pa * 2800 m^2

Calculating the expression above will give you the magnitude of the total force acting on the surface of the reservoir dam in a completely filled reservoir.

To find the magnitude of the total force acting on the vertical surface of the reservoir dam, we need to calculate the average pressure exerted by the water on that surface.

Let's start by finding the average pressure exerted at the midpoint of the vertical surface. We can assume that the pressure exerted by a fluid varies linearly with depth.

The midpoint of the vertical surface is at a height of 7 meters from the bottom.

The pressure at the water's surface is atmospheric pressure, which is approximately 1 atmosphere (ATM).

Using the linear variation of pressure with depth, we can calculate the pressure at the midpoint:

Pressure at midpoint = Atmospheric pressure + Change in pressure due to depth

The change in pressure due to depth can be calculated by multiplying the density of water (ρ) by the acceleration due to gravity (g) and the distance from the water's surface to the midpoint.

The density of water, ρ, is approximately 1000 kg/m³, and the acceleration due to gravity, g, is approximately 9.8 m/s².

Change in pressure due to depth = ρ * g * distance

In this case, the distance is the height from the midpoint to the water's surface, which is 7 meters.

Now, we can substitute the known values and calculate the change in pressure due to depth:

Change in pressure due to depth = 1000 kg/m³ * 9.8 m/s² * 7 m

= 68,600 N/m²

Now, we can calculate the pressure at the midpoint:

Pressure at midpoint = 1 ATM + Change in pressure due to depth

= 1 ATM + 68,600 N/m²

= 101,325 N/m² + 68,600 N/m²

= 169,925 N/m²

The area of the vertical surface of the dam is given as 200 m wide and 14 m high, so the magnitude of the total force can be calculated by multiplying the pressure at the midpoint by the area:

Total force = Pressure at midpoint * Area

Total force = 169,925 N/m² * (200 m * 14 m)

= 169,925 N/m² * 2,800 m²

= 475,790,000 N

Therefore, the magnitude of the total force acting on the surface of the reservoir dam is approximately 475,790,000 Newtons, when it is completely filled with water.