Pressure

Lying on a bed of nails — somehow — doesn’t hurt. Hundreds of nail tips share your weight, so each one bears only a tiny fraction. Now stand on a single nail. Same total weight, all concentrated on one tiny area. Excruciating.

The number that captures this difference is pressure: force divided by the area over which the force acts. Same force can be gentle (spread over big area, low pressure) or devastating (focused on a tiny point, high pressure). Snowshoes use this principle to keep you on top of snow; stiletto heels use it to leave dents in hardwood floors. Both are about pressure, not weight.

Defining Pressure

Pressure is a scalar, not a vector. At any point in a fluid, pressure pushes equally in every direction — up, down, sideways, all the same. This is fundamentally different from force, which has a specific direction. A force pushes; pressure squeezes.

Pressure Units

The MCAT uses four pressure units interchangeably. You have to be fluent in all of them.

Key relationships:

• 1 atm = 101,325 Pa ≈ $10^5$ Pa (use the approximation on the MCAT — much faster).
• 1 atm = 760 mmHg = 760 torr (mmHg and torr are exactly the same thing, different name).
• 1 atm ≈ 14.7 psi (rarely tested, but might appear in a passage).

Atmospheric Pressure

The atmosphere is a column of air about 100 km tall. That air has weight, and it presses down on everything below it. At sea level, that pressure is 1 atm = 101,325 Pa.

Atmospheric pressure decreases with altitude because there’s less air above you the higher you go:

• Sea level: 1 atm.
• Denver (~1.6 km): 0.83 atm.
• Top of Mt. Everest (~8.8 km): ~0.33 atm — about a third of sea-level pressure.
• Cruising altitude of a commercial plane (~10 km): ~0.25 atm (which is why cabins are pressurized — you’d black out otherwise).

This is why hikers feel short of breath at high altitudes: each lungful contains ~⅓ as many oxygen molecules at the top of Everest as at sea level.

Gauge Pressure vs. Absolute Pressure

This distinction trips up many students. There are two ways to report pressure:

• Absolute pressure is the total pressure at a point, including the atmospheric pressure pushing on everything. Always positive.
• Gauge pressure is the pressure above (or below) atmospheric pressure. A flat tire reads 0 on a tire gauge, but the air inside is still at 1 atm — gauge zero just means “same as atmospheric.”

Blood pressure readings ($\frac{120}{80}$ mmHg) are gauge pressures. They tell you how much the blood pressure exceeds atmospheric, not the absolute pressure in your arteries (which is actually 120 + 760 = 880 mmHg during systole).

Pressure in Fluids vs. Solids

In a solid, force can act in a specific direction over a specific area — a nail pressing into wood, a hammer striking a chisel. In a fluid, pressure acts equally in every direction at any given point.

That’s a direct consequence of fluids being unable to resist shear stress: if the pressure in a fluid were higher in one direction than another, the fluid would simply flow until the imbalance disappeared. So at every microscopic point, the fluid is squeezed equally from all sides.

Worked Example

A 70 kg person stands on one foot. The bottom of their shoe has area 0.02 m². What pressure does their foot exert on the floor? ($g = 10$ m/s².)

• Force = $mg = 700$ N.
• Pressure = $F/A = 700/0.02 = 35{,}000$ Pa = 35 kPa.

That’s about a third of an atmosphere — substantial, but spread over enough area that the floor doesn’t notice. Now switch to a stiletto heel with area 0.0001 m² instead: pressure = $700/0.0001 = 7{,}000{,}000$ Pa = 7 MPa. That’s why stilettos make dents and regular shoes don’t.