A circular observation window, 6 feet in diameter, is to be placed near the bottom of a pool 10 feet deep. Calculate the liquid pressure on the window.
Hint: Density of water $=62.4 lb / ft ^3$.
Round your answer to the nearest pound.
Answer (2)
Calculate the radius of the circular window: r = 2 6 = 3 feet.
Calculate the area of the circular window: A = π ( 3 2 ) = 9 π square feet.
Calculate the pressure at the bottom of the pool: P = 62.4 × 10 = 624 lb/ft^2.
Calculate the force on the window: F = 624 × 9 π ≈ 17643 lb. The liquid pressure on the window is 17643 lb.
Explanation
Problem Setup We are asked to calculate the liquid pressure on a circular window placed at the bottom of a pool. We are given the diameter of the window, the depth of the pool, and the density of water.
Calculate the Radius First, we need to find the radius of the circular window. The radius is half of the diameter, so: r = 2 d = 2 6 = 3 feet
Calculate the Area Next, we calculate the area of the circular window using the formula: A = π r 2 = π ( 3 2 ) = 9 π ft 2
Calculate the Pressure The pressure at a certain depth in a liquid is given by: P = ρ g h where ρ is the density of the liquid, g is the acceleration due to gravity, and h is the depth. Since we are given the density of water as 62.4 lb/ft 3 and the depth as 10 feet , we can calculate the pressure: P = 62.4 × 10 = 624 lb/ft 2
Calculate the Force Now, we can calculate the force on the window by multiplying the pressure by the area: F = P × A = 624 × 9 π = 5616 π ≈ 17642.67 lb
Round the Force Finally, we round the force to the nearest pound: F ≈ 17643 lb
Examples
Understanding liquid pressure is crucial in various real-world applications. For instance, when designing submarines or underwater habitats, engineers must accurately calculate the pressure exerted by water at different depths to ensure the structural integrity of the vessel. Similarly, in civil engineering, calculating the hydrostatic pressure on dams and retaining walls is essential for safe and stable construction. This problem demonstrates a fundamental principle in fluid mechanics that helps engineers create safe and efficient designs for underwater structures.
The liquid pressure on the observation window is approximately 17643 lb, calculated using the depth of the pool and the area of the circular window. This pressure is derived from the density of water and the area of the window. The formula for calculating pressure in a liquid helps us understand the forces acting on submerged objects.
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