The bottled liquid (sp gr = 0.9) in the fig. is under pressure, as shown by the manometer reading. Compute the net force on the 50mm radius concavity in the bottom of the bottle.

Net Force on Concavity

Problem Statement

The bottled liquid (specific gravity = 0.9) in the figure is under pressure, as shown by the manometer reading. Compute the net force on the 50 mm radius concavity at the bottom of the bottle.

Solution

1. Symmetry Consideration

From symmetry, the horizontal component of the force \( F_H \) is zero:

\( F_H = 0 \)

2. Manometric Equation for Pressure

Using the manometric equation:

\( P_{AA} + \gamma \cdot 0.07 = \gamma_{Hg} \cdot 0.12 \)

Substituting values:

\( \gamma = 0.9 \cdot 9810 \, \text{N/m}^3 \)
\( \gamma_{Hg} = 13.6 \cdot 9810 \, \text{N/m}^3 \)

\( P_{AA} + 0.9 \cdot 9810 \cdot 0.07 = 13.6 \cdot 9810 \cdot 0.12 \)

Simplifying:

\( P_{AA} = 15392 \, \text{N/m}^2 \)

3. Net Vertical Force \( F_V \)

The vertical force is the sum of the pressure force and the weight of the liquid below \( AA \):

\( F_V = P_{AA} \cdot A_{\text{bottom}} + \gamma \cdot \text{Volume}_{\text{below AA}} \)

Substituting:

\( \text{Volume}_{\text{below AA}} = \text{Volume of cylinder of height 16 cm} \)
\( – \text{Volume of hemisphere of radius 50 mm} \)

Calculating the individual terms:

\( A_{\text{bottom}} = \pi \cdot (0.05)^2 \)
\( \text{Volume of cylinder} = \pi \cdot (0.05)^2 \cdot 0.16 \)
\( \text{Volume of hemisphere} = \frac{1}{2} \cdot \frac{4}{3} \cdot \pi \cdot (0.05)^3 \)

Substituting these into the equation:

\( F_V = 15392 \cdot \pi \cdot (0.05)^2 + 0.9 \cdot 9810 \cdot \left[ \pi \cdot (0.05)^2 \cdot 0.16 \right. \)
\( \left. – \frac{1}{2} \cdot \frac{4}{3} \cdot \pi \cdot (0.05)^3 \right] \)

Simplifying:

\( F_V = 129.7 \, \text{N (down)} \)

Result:

Net vertical force: \( F_V = 129.7 \, \text{N (down)} \)

Explanation

Pressure Calculation: The pressure at point \( AA \) is determined using the manometric equation, which considers the heights of the liquid columns and their specific gravities.
Force Components: The horizontal force is zero due to symmetry, while the vertical force includes the contribution from the pressure on the bottom surface and the weight of the liquid.
Volume Calculation: The volume of liquid below \( AA \) is computed by subtracting the volume of the hemisphere from the volume of the cylinder.

Physical Meaning

This problem demonstrates the application of hydrostatics to compute forces on a concavity at the bottom of a bottle. It highlights the following:

Symmetry: Horizontal forces cancel out due to symmetry, leaving only vertical forces to consider.
Manometric Principles: The pressure difference is calculated using the heights and specific gravities of the liquids in the manometer.
Structural Design: The forces computed are essential for understanding the stress and pressure distribution on container surfaces, aiding in structural design.

This type of analysis is crucial in the design of vessels and containers to ensure safety and stability under fluid pressure.