To find the hydrostatic pressure, hydrostatic force on the bottom of the aquarium, and the hydrostatic force on one end of the aquarium, we will use the following formulas:
-
Hydrostatic Pressure (P): \[ P = \rho g h \] where:
- \(P\) = pressure (in pascals)
- \(\rho\) = density of water (approximately \(1000 , \text{kg/m}^3\))
- \(g\) = acceleration due to gravity (approximately \(9.81 , \text{m/s}^2\))
- \(h\) = depth of the fluid (in meters)
-
Hydrostatic Force (F): \[ F = P \cdot A \] where:
- \(F\) = force (in newtons)
- \(A\) = area (in square meters)
Step 1: Hydrostatic Pressure on the Bottom of the Aquarium
The depth of the water \(h\) is \(3 , \text{ft}\). We need to convert this to meters: \[ h = 3 , \text{ft} \times \frac{0.3048 , \text{m}}{1 , \text{ft}} \approx 0.9144 , \text{m} \]
Now, substituting in the values to calculate the hydrostatic pressure: \[ P = \rho g h = 1000 , \text{kg/m}^3 \times 9.81 , \text{m/s}^2 \times 0.9144 , \text{m} \approx 8985.5 , \text{Pa} , (\text{or} , \text{N/m}^2) \]
Step 2: Hydrostatic Force on the Bottom of the Aquarium
The area of the bottom of the aquarium \(A\) is given by: \[ A = \text{length} \times \text{width} = 5 , \text{ft} \times 2 , \text{ft} = 10 , \text{ft}^2 \]
Convert the area into square meters: \[ A = 10 , \text{ft}^2 \times \left( \frac{0.3048 , \text{m}}{1 , \text{ft}} \right)^2 = 10 \times 0.092903 , \text{m}^2 \approx 0.92903 , \text{m}^2 \]
Now, we can calculate the hydrostatic force on the bottom: \[ F = P \cdot A \approx 8985.5 , \text{Pa} \times 0.92903 , \text{m}^2 \approx 8346.05 , \text{N} \]
Step 3: Hydrostatic Force on One End of the Aquarium
The area of one end of the aquarium (which is a rectangle) is: \[ A_{\text{end}} = \text{width} \times \text{depth} = 2 , \text{ft} \times 3 , \text{ft} = 6 , \text{ft}^2 \] Convert this to square meters: \[ A_{\text{end}} = 6 , \text{ft}^2 \times 0.092903 , \text{m}^2 \approx 0.557418 , \text{m}^2 \]
To find the hydrostatic force on one end, we need to calculate the average pressure on that end. The average depth is half the total depth: \[ \text{Average depth} = \frac{h}{2} = \frac{0.9144}{2} \approx 0.4572 , \text{m} \] Now we can calculate the average pressure: \[ P_{\text{avg}} = \rho g \cdot \text{Average depth} = 1000 , \text{kg/m}^3 \times 9.81 , \text{m/s}^2 \times 0.4572 , \text{m} \approx 4486.375 , \text{Pa} \]
Now, we can calculate the force on one end: \[ F_{\text{end}} = P_{\text{avg}} \cdot A_{\text{end}} \approx 4486.375 , \text{Pa} \times 0.557418 , \text{m}^2 \approx 2505.49 , \text{N} \]
Summary of Results:
- Hydrostatic Pressure on the Bottom: \( P \approx 8985.5 , \text{Pa} \)
- Hydrostatic Force on the Bottom: \( F \approx 8346.05 , \text{N} \)
- Hydrostatic Force on One End: \( F_{\text{end}} \approx 2505.49 , \text{N} \)