Q.Explain why

1. Key idea: Pascal’s principle — pressure in a fluid increases with depth due to the weight of the fluid column above. Blood is a fluid. The vertical height of the blood column from the heart to the feet is greater than that to the brain. Hence, the hydrostatic pressure $P = \rho g h$ is larger at the feet.
2. Key idea: Atmospheric density is not uniform — it decreases exponentially with height. The atmosphere is compressible; most of its mass lies in the first few kilometres. At $6\ \text{km}$, the air density is already much lower, so the weight of the column above is halved. Beyond that, the remaining $94\ \text{km}$ contributes very little because the density is extremely low.
3. Key idea: Pressure is a scalar because it acts equally in all directions at a point, regardless of the orientation of the area. Although $P = F/A$, the force is always normal to the surface. At any point in a static fluid, the magnitude of this normal force per unit area is the same in every direction — that directional independence makes it a scalar.
   ✓Final answer

   1. Blood pressure is greater at the feet due to the larger height of the blood column.
   2. Atmospheric pressure halves by $6\ \text{km}$ because air density drops rapidly, so most of the atmosphere’s weight lies below that height.
   3. Hydrostatic pressure is a scalar because it acts uniformly in all directions at a point.

(a) Blood pressure at the feet vs. at the brain

The concept: In a fluid at rest, pressure at any point is given by $P = P_0 + \rho g h$, where $h$ is the vertical depth below the reference level (usually the top surface of the fluid). For the human body, the heart is the pump, but once blood is in the arteries, gravity acts on the blood column.

Step-by-step reasoning:

1. Choose a reference level. The heart is the pump that generates the driving pressure. But for the static component of blood pressure (the part due to gravity), we can think of the heart as the "source" at roughly the same height as the brain when standing.

2. Compare the vertical columns. When a person stands upright:

   • The brain is about $0.4\ \text{m}$ above the heart.
   • The feet are about $1.3\ \text{m}$ below the heart. So the vertical height of the blood column above the feet is much larger than the height of the blood column above the brain.

3. Apply the hydrostatic formula. Let $P_{\text{heart}}$ be the pressure at heart level. Then:

   • At the brain: $P_{\text{brain}} = P_{\text{heart}} - \rho g h_{\text{up}}$, where $h_{\text{up}} \approx 0.4\ \text{m}$.
   • At the feet: $P_{\text{feet}} = P_{\text{heart}} + \rho g h_{\text{down}}$, where $h_{\text{down}} \approx 1.3\ \text{m}$.

   Since $h_{\text{down}} > h_{\text{up}}$, we get $P_{\text{feet}} > P_{\text{brain}}$.

4. Numerical sense. With $\rho_{\text{blood}} \approx 1.06 \times 10^3\ \text{kg/m}^3$ and $g \approx 9.8\ \text{m/s}^2$, the difference is roughly $\rho g (h_{\text{down}} + h_{\text{up}}) \approx 1.06 \times 10^3 \times 9.8 \times 1.7 \approx 1.77 \times 10^4\ \text{Pa}$, or about $133\ \text{mm Hg}$. That's a substantial difference — which is why blood pressure readings are always taken at heart level for consistency.

(b) Atmospheric pressure drop at 6 km vs. total height of atmosphere

The concept: Unlike water, air is highly compressible. Its density decreases exponentially with height. So most of the atmosphere's mass lies in the lowest few kilometres.

Step-by-step reasoning:

1. Recall the hydrostatic equation for a compressible fluid. For any fluid at rest, $dP = -\rho g \, dh$. For an incompressible fluid (like water), $\rho$ is constant, so $P$ decreases linearly with height. For air, $\rho$ itself depends on $P$ (and temperature).

2. The exponential approximation. If we assume an isothermal atmosphere (constant temperature), the ideal gas law gives $\rho = \frac{PM}{RT}$, where $M$ is molar mass. Substituting into $dP = -\rho g \, dh$ yields:

Integrating from sea level ($h=0$, $P=P_0$) to height $h$:

where $H = \frac{RT}{Mg}$ is the scale height — the height over which pressure falls by a factor of $e$.

3. Plug in numbers. For Earth's atmosphere, $T \approx 288\ \text{K}$ (average), $M \approx 0.029\ \text{kg/mol}$, $g \approx 9.8\ \text{m/s}^2$, $R = 8.314\ \text{J/(mol·K)}$:

4. Check at 6 km. At $h = 6\ \text{km}$:

That's nearly half — exactly as the question states.

5. Why does the atmosphere extend beyond 100 km? The exponential decay means pressure never truly reaches zero; it just gets vanishingly small. At $h = 100\ \text{km}$, $P/P_0 = e^{-100/8.4} \approx 10^{-5}$, which is essentially vacuum for practical purposes. So the "top" of the atmosphere is fuzzy — the tail is long, but the bulk of the mass (and hence the pressure) is concentrated in the first scale height.

(c) Hydrostatic pressure is a scalar, even though pressure = force/area

The concept: The definition $P = F/A$ involves force (a vector) and area (which has orientation), so one might think pressure has a direction. But in a fluid at rest, the force on any surface element is always perpendicular to that surface, and its magnitude is the same regardless of the surface's orientation.

Step-by-step reasoning:

1. Pressure is defined at a point. Consider an infinitesimally small area $\Delta A$ inside a fluid. The fluid on one side exerts a force $\Delta \vec{F}$ on the fluid on the other side. For a fluid at rest, this force is always normal to the surface — there is no shear component.

2. Pascal's principle. The magnitude of this normal force per unit area, $\Delta F / \Delta A$, is the same no matter how you orient the area element. That is, if you rotate the tiny area, the force magnitude changes proportionally so that the ratio stays constant. This is a consequence of the fluid being unable to support shear stress.

3. Why it's a scalar. A scalar is a quantity that has magnitude but no direction. Pressure has magnitude (the value $P$ at that point) but does not point in any particular direction. The force on a surface is $\vec{F} = P \Delta A \, \hat{n}$, where $\hat{n}$ is the unit normal to the surface — the direction comes from the surface, not from the pressure itself. Pressure is like temperature: it has a value at each point, but it doesn't "point" anywhere.

4. Contrast with force. Force is a vector because it has both magnitude and a specific direction. Pressure, being the isotropic component of the stress tensor, is the same in all directions. In fact, the full stress tensor for a fluid at rest is $P$ times the identity matrix — all diagonal entries equal, off-diagonals zero.

1. blood column height above feet > above brain => P_feet>P_brain via P=P0+rhogh.
2. air density depends on pressure (ideal gas) => exponential decay P=P0*exp(-h/H), H~=8.4km; at h=6km, P/P0~=0.49.
3. static fluid can't sustain shear => force per area same in all directions -- pressure is a scalar, direction comes from multiplying by the surface normal.