Hydrostatic Gauge Pressure Calculation:

• Water Depth Determination:
• Total pond depth = $8\text{ m}$
• Filled portion = $3/4^\text{th}$ of $8\text{ m} = \frac{3}{4} \times 8 = 6\text{ m}$
• Swimmer Position:
• Swimmer is located at $2\text{ m}$ above the pond bottom.
• Submergence depth ($h$) below the free water surface = Total water depth - Height above bottom = $6\text{ m} - 2\text{ m} = 4\text{ m} = 400\text{ cm}$.
• Pressure Calculation:
  In gravitational metric units ($\text{g/cm}^2$):
  $$P = \rho \times h = 1\text{ g/cm}^3 \times 400\text{ cm} = 400\text{ g/cm}^2$$

Physical Principle of Droplet Sphericity:

• Surface Free Energy Minimization: Molecules at the liquid-air interface experience an unbalanced net inward cohesive attraction toward the bulk liquid, generating surface tension ($\sigma$).
• Geometric Optimum: Under the action of surface tension, the liquid droplet spontaneously minimizes its surface free energy by minimizing its surface area for a given enclosed volume. For any geometric shape, a sphere possesses the minimum surface area per unit volume.

Standard Relationships in Fluid Pressure Measurement:

• Fundamental Equations:
• Absolute pressure is measured relative to absolute zero pressure (perfect vacuum):
  $$P_{\text{abs}} = P_{\text{atm}} + P_{\text{gage}}$$
  Rearranging for gage pressure:
  $$P_{\text{gage}} = P_{\text{abs}} - P_{\text{atm}}$$

Principles of Machines (Mechanical Advantage vs Velocity Ratio):

• Definitions:
• Mechanical Advantage (MA) = $\frac{\text{Load } (W)}{\text{Effort } (P)}$
• Velocity Ratio (VR) = $\frac{\text{Distance moved by Effort } (y)}{\text{Distance moved by Load } (x)}$
• Efficiency ($\eta$) = $\frac{\text{MA}}{\text{VR}}$
• Machine Realities: In any real/actual machine, friction at moving contacts always consumes a portion of input work, so efficiency is strictly less than $100\%$ ($\eta \lt 1.0$):
  $$\frac{\text{MA}}{\text{VR}} \lt 1 \implies \text{MA} \lt \text{VR}$$