please help!! A storage tank 32 m high is filled with pure water. (Assume the tank is open and exposed to the atmosphere at the top.)
(a) Find the gauge pressure at the bottom of the tank.
(b) Calculate the magnitude of the net force that acts on a square access hatch at the bottom of the tank that measures 0.9 m by 0.9 m.
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Sure, I can help you with that. Let's go step by step!
(a) To find the gauge pressure at the bottom of the tank, we can use the formula for pressure:
Pressure = Density * gravitational acceleration * height
Here, the density of water is approximately 1000 kg/m³, and the gravitational acceleration is 9.8 m/s². The height of the tank is given as 32 m.
So, the gauge pressure at the bottom of the tank is:
Pressure = 1000 kg/m³ * 9.8 m/s² * 32 m
Calculating this, we get:
Pressure = 313,600 Pascal (Pa)
(b) To calculate the magnitude of the net force acting on the square access hatch at the bottom of the tank, we need to consider the pressure acting on the hatch.
Force = Pressure * Area
Given that the area of the hatch is 0.9 m by 0.9 m, the total area is:
Area = 0.9 m * 0.9 m
Calculating this, we get:
Area = 0.81 square meters (m²)
Now, we can calculate the force:
Force = 313,600 Pa * 0.81 m²
Calculating this, we get:
Force = 253,536 Newtons (N)
So, the magnitude of the net force acting on the square access hatch at the bottom of the tank is 253,536 N.
To find the answers to both parts (a) and (b), we need to understand a few concepts:
1. Gauge pressure: This is the pressure measured relative to atmospheric pressure. In this case, the tank is open to the atmosphere, so the gauge pressure at the bottom is the pressure that the water exerts on the bottom of the tank.
2. Hydrostatic pressure: It is the pressure exerted by a fluid due to the weight of the fluid above it. The hydrostatic pressure depends on the density of the fluid and the height of the column of fluid.
Now let's calculate the answers step by step:
(a) Finding the gauge pressure at the bottom of the tank:
The pressure at the bottom can be calculated using the equation: P = ρgh, where P is the pressure, ρ is the density of water, g is the acceleration due to gravity, and h is the height of the column of water.
Given:
Height of the tank (h) = 32 m
Density of water (ρ) = 1000 kg/m³ (approximate value)
Acceleration due to gravity (g) = 9.8 m/s² (approximate value)
Substituting these values into the equation:
P = (1000 kg/m³) * (9.8 m/s²) * (32 m) = 313,600 Pa
Therefore, the gauge pressure at the bottom of the tank is 313,600 Pa.
(b) Calculating the magnitude of the net force on the access hatch:
The force acting on the square access hatch at the bottom is equal to the pressure exerted on it multiplied by its area.
Given:
Dimensions of the hatch (length and width) = 0.9 m
Area of the hatch = (0.9 m) * (0.9 m) = 0.81 m²
Using the gauge pressure obtained in part (a):
Force = Pressure * Area = 313,600 Pa * 0.81 m² = 254,256 N
Therefore, the magnitude of the net force acting on the access hatch is 254,256 Newtons.
Please note that in both parts, it is assumed that the pressure is uniform throughout the base of the tank and the access hatch.