Question

Explain why 

(a) The blood pressure in humans is greater at the feet than at the brain 

(b) Atmospheric pressure at a height of about 6 km decreases to nearly half of its value at the sea level, though the height of the atmosphere is more than 100 km 

(c) Hydrostatic pressure is a scalar quantity even though pressure is force divided by area.

Answer 1

(a) Blood Pressure: Feet > Brain

Hydrostatic pressure increases with depth

Standing human (~1.7 m tall):

$$\Delta P = \rho g h$$ $$P_\text{feet} = P_\text{heart} + \rho g (1.2 \, \text{m})$$ $$P_\text{brain} = P_\text{heart} - \rho g (0.5 \, \text{m})$$

Feet are 1.7 m below brain → higher column of blood → greater pressure

(b) Atmospheric Pressure Halves at 6 km

Exponential decrease due to density falloff

$$P(h) = P_0 e^{-h/H}, \quad H \approx 8 \, \text{km (scale height)}$$

At 6 km: \(e^{-6/8} = e^{-0.75} \approx 0.47\)

• 90% of atmosphere's mass in first 16 km
• Pressure drops rapidly despite 100+ km total height
• Less air mass above → less weight → lower pressure

(c) Hydrostatic Pressure is Scalar

Random molecular collisions from all directions

• Force/area → should be tensor (directional)
• But: Isotropic fluid → equal pressure every direction
• Molecular motion random → net force omnidirectional

$$\vec{F}_\text{wall} = P \cdot A \cdot \hat{n}$$ $$P = |\vec{F}|/A \text{ (magnitude only, no direction)}$$

Pressure = magnitude of force per area, independent of surface orientation

Summary Table

Phenomenon	Key Principle	Result
(a) Blood pressure	\(P = \rho g h\)	Feet > Brain
(b) Air pressure	Exponential \(e^{-h/H}\)	½ at 6 km
(c) Scalar pressure	Isotropic molecular motion	Direction independent