IP Physics Notes (Upper Secondary, Year 3-4): 5) Pressure

Quick recap -- Pressure concentrates force over area. In fluids it scales with depth, density, and gravity; in gas systems it trades off with volume when temperature stays fixed.

Defining Pressure

• Pressure is force acting perpendicular to a surface per unit area.
• Formula: \[ P = \frac{F}{A} \]
  • \( P \) in \( \pu{Pa} = \pu{N.m-2} \)
  • \( F \) in \( \pu{N} \)
  • \( A \) in \( \pu{m2} \)
• Increase pressure by raising the normal force or shrinking the contact area. Decrease it by spreading the force over a wider area.

Pressure in Solids: Force vs Area

• When the contact is angled, resolve the force to the component perpendicular to the surface before using \( P = \frac{F}{A} \).
• Engineers tweak both force and area (e.g., tyre width, footprints of machine supports) to keep ground pressure within safe limits.

Hydrostatic Pressure

• Fluids transmit pressure equally in all directions; deeper layers carry the weight of fluid above.
• Pressure at depth \( h \) in a fluid of density \( \rho \): \[ P = \rho g h \]
• Derivation: take a column of cross-sectional area \( A \) and height \( h \). Its weight is \( W = \rho A h g \). Using \( P = \frac{F}{A} \), the pressure on the base equals \( \frac{W}{A} = \rho g h \).
• Hydrostatic pressure depends only on depth, density, and gravity -- not on container shape.
• Applied cases: dam walls thicken near the base, submarines need hulls that withstand higher \( \rho g h \) at depth.

Worked Example: Swimming Pool Wall

At ( \pu{3.0 m} ) depth, water exerts pressure \[ P = \rho g h = \pu{1000 kg.m-3} \times \pu{9.81 m.s-2} \times \pu{3.0 m} = \pu{2.94 \times 10^{4} Pa} \] Each square metre of wall must therefore resist about ( \pu{29.4 kN} ) pushing sideways.

Pascal's Principle & Hydraulic Systems

• Pascal's principle: a pressure change applied to an enclosed, incompressible fluid transmits undiminished to every part of the fluid.
• In a hydraulic press with pistons of area \( A_1 \) and \( A_2 \): \[ P = \frac{F_1}{A_1} = \frac{F_2}{A_2} \]
• Force multiplication: \[ F_2 = F_1 \times \frac{A_2}{A_1} \]
• Trade-off: the larger output force moves a shorter distance so overall energy is conserved. Hydraulic brakes, car jacks, and presses all rely on this ratio.

Worked Example: Service Garage Jack

A mechanic pushes with ( \pu{180 N} ) on a ( \pu{5.0 cm2} ) input piston. The output piston area is ( \pu{150 cm2} ). \[ F_2 = 180 \times \frac{150}{5.0} = \pu{5400 N} \] The car experiences a lifting force of about ( \pu{5.4 kN} ).

Measuring Pressure

• Barometer: a sealed mercury column. Atmospheric pressure equals the hydrostatic pressure of the mercury column: \( P_\text{atm} = \rho_\text{Hg} g h \). Standard sea-level pressure ( \pu{1.013 \times 10^5 Pa} ) supports ( \pu{760 mm} ) of mercury.
• Manometer: U-tube with one side connected to the gas source.
  • Closed-end: pressure equals \( \rho g h \) of the column difference.
  • Open-end: difference between gas and atmospheric pressure given by the column height difference.
• Digital sensors convert diaphragm deformation into electrical signals -- but conceptually they still report force per area.

Gas Pressure & Boyle's Law

• For a fixed mass of gas at constant temperature, pressure is inversely proportional to volume: \[ P_1 V_1 = P_2 V_2 \]
• Doubling pressure halves volume; halving pressure doubles volume.
• Classroom examples:
  • Syringe: pulling the plunger increases volume, lowering pressure so fluid/air enters.
  • Breathing: diaphragmatic movement changes lung volume, producing pressure gradients with the atmosphere.
  • Scuba ascent: decreasing external pressure lets lung volume expand; divers ascend slowly so the gas escapes instead of over-expanding tissues.

Key Takeaways

• Start every problem by identifying the relevant area, depth, or volume change, then apply \( P = \frac{F}{A} \) or \( P = \rho g h \).
• Pascal's principle enables hydraulic multipliers but keeps the pressure equal throughout the fluid.
• Pressure measurement tools compare unknown pressure to a known fluid column height.
• Gas pressure trades with volume when temperature is fixed -- keep \( P V \) constant and track unit conversions carefully.