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The image shows a pre-test question about determining the total force on a panel (AB) in a tank that contains water and is pressurized to 80 cmHg. Here is the analysis and solution breakdown:

### Given Data:
- Water tank is pressurized to **80 cmHg**.
- Dimensions of panel **AB**:
  - Width of the tank: **2 m**
  - Height of panel AB: **3 m** (from point A to B)
  - Total height of the tank: **4 m** (from bottom to top)
  - Top of the tank is 2 meters above point B.
  
We need to:
1. **Determine the total force per meter width on panel AB**.
2. **Determine the value and location of the total force on panel AB** if the width of the tank is 2 meters.

### Assumptions:
- The tank is filled with water, which has a density (ρ) of approximately **1000 kg/m³**.
- Gravitational acceleration (g) is **9.81 m/s²**.

### Part A: Total Force per Meter Width on Panel AB

The total force exerted by a liquid on a submerged surface is calculated by integrating the pressure over the surface area. For a vertical panel submerged in water, the pressure at a given depth (h) is given by:

\[
P(h) = P_0 + \rho g h
\]

Where:
- \( P_0 \) is the pressure at the surface, which is **80 cmHg** (convert this to Pascals),
- \( \rho \) is the density of water (**1000 kg/m³**),
- \( g \) is gravitational acceleration (**9.81 m/s²**),
- \( h \) is the depth below the surface.

1. **Convert pressure:**
   - 1 cmHg ≈ 1333.22 Pa, so:
     \[
     P_0 = 80 \, \text{cmHg} = 80 \times 1333.22 \, \text{Pa} = 106657.6 \, \text{Pa}
     \]

2. **Force on panel AB** (per meter width):
   The total force on panel AB is the integral of the pressure over its height (from point A at \( h = 3 \, \text{m} \) to point B at \( h = 0 \, \text{m} \)).

### Part B: Total Force for 2 meters width and its Location

Once the total force is determined for a width of 1 meter, it can be scaled for the given width of 2 meters. Additionally, the **location** of the force (center of pressure) is found using:

\[
h_{cp} = \frac{I_G + A \cdot h_c}{A \cdot P(h_c)}
\]

Where \( h_c \) is the centroid depth and \( I_G \) is the second moment of area.

---

Let me compute the results for these calculations and then present the solution.### Solution:

1. **Total Force per Meter Width on Panel AB:**
   - The force per meter width on panel AB is approximately **364,118 N/m**.

2. **Total Force for the 2-meter Width Tank:**
   - The total force on the panel AB, given the width of the tank is 2 meters, is approximately **728,236 N**.

3. **Location of the Total Force (Center of Pressure):**
   - The center of pressure (the point where the total force can be considered to act) is located approximately **1.56 meters** from the top of panel AB.

Would you like more detailed steps or explanations on any part of the solution?

---

Here are 5 related questions for further exploration:
1. How would the force change if the tank were filled with a different liquid, such as oil with a different density?
2. How is the center of pressure formula derived, and why does it differ from the centroid?
3. What effect would a higher pressurization, say 100 cmHg, have on the total force?
4. How does the pressure distribution vary with depth in a liquid, and why is it linear?
5. How would the solution change if the panel were inclined rather than vertical?

**Tip:** Always ensure to convert units properly when dealing with pressures, especially if they're given in unconventional units like cmHg or atm.

Water in a tank is pressurized to 80 cmHg. a) Determine the total force per meter width on panel AB, b) if the width of the tank is 2 m, what is the value and location of the total force on panel AB.

Learn how to calculate the total force per meter width and the total force on panel AB in a water tank pressurized to 80 cmHg. Includes a breakdown of fluid mechanics principles, hydrostatic pressure distribution, and center of pressure calculation. Suitable for engineering students.

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