You are working as a summer intern for the Illinois Department of Natural Resources (DNR) and are assigned to some initial work on water resources project. The department will be overseeing the construction of a dam to create a large fresh water lake that will be approximately 14 meters deep. A horizontal pipe 1.2 meters long and 3 cm in diameter will pass through the dam at a depth of 7 meters to allow for release of the water in emergencies and for sampling. In normal situations, a plug will secure the pipe opening. 1)What will be the total force on the left side of the plug?FLEFT = N2)What is the total force on the right side of the plug?FRIGHT = N3)What is the gauge pressure on the plug?Pgauge = Pa4)If the plug were to be removed, how much water would flow out in 3 hours??VOL = m3

1)   F = 71.6 N , 2)    F = 120.2 N , 3)  P_m=  68.600 Pa, 4)  V = 2.4210-5 m³

Explanation:

This is a problem of fluid mechanics, to find the force we must use its definition

         P = F / A

         F = P A

The area of the circular pipe is

         A = π r² = π d²/4

The pressure is given by the expression

         P = P_atm + ρ g h

1) the force on the outer side is

       P = P_atm

we substitute in the expression of force

         F = P_atm π d² / 4

let's calculate

         F = 1,013 10⁵ π 0.03²/4

         F = 7.16 10¹ N

         F = 71.6 N

2) the force on the inner side

  the pressure

       P = P_atm + ρ g h

       P = 1,015 10⁵ + 1,000 9.8 7

        P = 1,701 10⁵ Pa

        F = 1,701 10⁵ π 0.03² / 4

        F = 1,202 10²

        F = 120.2 N

3) manometric pressure

       Pm = ρ g h

       P m = 1000 9.8  7

       P_m=  68.600 Pa

4) In this part they ask for the volume that comes out in time t= 3 h

   to calculate this volume we can use the flow ratio

         Q = A v

          V t = A v

          V = A v / t

sent is the velocity of the water that comes out, to calculate it we use the Bernoulli equation

we will use index 1 for the lake surface and ionice 2 apa the position of the plug

           P₁ + ρ g v₁² + ρ g y₁ = P₂ + ρ g v₂² + ρ g y₂

As the lake has much more capacity than the pipeline, the velocity of the surface of the lake is peeling, in this case we approximate it steel

           (P₁-P₂) + ρ G (y₁ -y₂) = ρ g v₂²

           1000 9.8 v₂² = ρ g h + 1000 9.8 (7-0)

             9800 v₂² = 1000 9.8 7 + 68600

              v₂ = √ (137200)

               v₂ = 370.4 m / s

             t = 3 h (3600s / h) = 10800 s

we substitute in the volume equation

             V = π d²/4   370.4 / 10800

             V = 2.4210-5 m³

more than, less

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