Pressure and Its Applications

1. What is Pressure?

Think of pressure as the “push” a fluid gives per unit area. The average pressure on a flat surface is  \( P_{\text{av}} = \dfrac{F}{A} \). For an infinitesimally small area, we write  \( P = \displaystyle \lim_{\Delta A \to 0} \dfrac{\Delta F}{\Delta A} \). Pressure is a scalar (no direction) and its SI unit is the pascal (Pa). One atmosphere (1 atm) equals \(1.013 \times 10^{5}\,\text{Pa}\). :contentReference[oaicite:0]{index=0}

2. Density & Relative Density

Density tells us how “packed” matter is: \( \rho = \dfrac{m}{V} \). Water at 4 °C has \( \rho = 1.0 \times 10^{3}\,\text{kg m}^{-3} \). Relative density is the ratio of any substance’s density to this value (dimensionless). :contentReference[oaicite:1]{index=1}

3. Pascal’s Law

When you squeeze a fluid, it sends that extra pressure everywhere equally. So points at the same height feel the same pressure, and an increase applied at one spot travels unchanged throughout the fluid. :contentReference[oaicite:2]{index=2}

4. How Pressure Changes with Depth

For two points separated vertically by \(h\): \( P_2 – P_1 = \rho g h \). If point 1 sits at the free surface open to the atmosphere, the absolute pressure at depth \(h\) becomes \( P = P_a + \rho g h \). The term \( \rho g h \) is the gauge pressure—the “extra” pressure you feel below the surface. :contentReference[oaicite:3]{index=3}

Hydrostatic Paradox

Shape does not matter—only depth. Connected vessels of any shape hold liquid to the same height because pressure at the bottom is identical. :contentReference[oaicite:4]{index=4}

5. Atmospheric Pressure & Measuring Devices

  • Mercury barometer: A column of mercury stands about 76 cm tall at sea level, giving \( P_a = \rho g h \). Units you’ll meet: mm Hg (or torr) and the bar (1 bar = 105 Pa). :contentReference[oaicite:5]{index=5}
  • Open-tube manometer: Compares an unknown pressure to atmospheric pressure; the height difference of the liquid column tells you \( P – P_a = \rho g h \). :contentReference[oaicite:6]{index=6}

6. Hydraulic Machines – Big Force from a Small Push

Because fluids transmit pressure unchanged, a small piston (area \(A_1\)) pushing with force \(F_1\) creates pressure \(P = \dfrac{F_1}{A_1}\). The same pressure acts on a large piston (area \(A_2\)), producing  \( F_2 = P A_2 = F_1 \dfrac{A_2}{A_1} \). This is the secret behind hydraulic lifts and brakes. :contentReference[oaicite:7]{index=7}

7. Quick Numerical Feel

  • Standing on femurs: A 40 kg upper body supported by two thigh bones (total area 20 cm²) feels about \(2 \times 10^{5}\,\text{Pa}\) on the bones. :contentReference[oaicite:8]{index=8}
  • Diving 10 m: Pressure doubles—roughly 2 atm at that depth in fresh water. :contentReference[oaicite:9]{index=9}
  • Submarine at 1 km: Gauge pressure ≈ \(10^{3}\,\text{atm}\); windows must withstand forces of hundreds of kilonewtons. :contentReference[oaicite:10]{index=10}
  • Hydraulic lift example: A 10 N push on a 1 cm-diameter piston can raise a 90 N load on a 3 cm-diameter piston. :contentReference[oaicite:11]{index=11}

High-Yield Ideas for NEET

  1. Depth formula \( P = P_a + \rho g h \) and its “double-at-10 m” trick for water-based questions.
  2. Pascal’s law applications: hydraulic lift, hydraulic brakes—expect ratio problems using \( F_2 = F_1 \frac{A_2}{A_1} \).
  3. Mercury barometer mechanics, unit conversions (mm Hg ↔ Pa ↔ atm).
  4. Difference between absolute and gauge pressure; manometer readings.
  5. Density and relative density definitions, especially comparing fluids (water vs. mercury) in barometers or manometers.