A tall, narrow cylinder and a wide, shallow bowl are both filled with water to a depth of exactly 0.5 meters. How does the pressure at the bottom of the cylinder compare to the pressure at the bottom of the bowl?
AThe cylinder has higher pressure because it contains more water pressing down per unit area
BThe bowl has higher pressure because the wider base spreads force over more area
CThey are equal — pressure depends only on depth and fluid density, not container shape
DThe cylinder has higher pressure because the narrow walls cannot support the water weight
This is the hydrostatic paradox: pressure at the bottom of a fluid depends only on the vertical depth h and the fluid density ρ, giving P = P₀ + ρgh. Container shape and total fluid volume are irrelevant. At 0.5 m depth with the same fluid (water), both containers have identical bottom pressure. Option A is the classic misconception — intuitively, more water seems like more pressure, but the force per unit area is set entirely by depth. The force over a larger area is larger, but force per unit area (pressure) is the same.
Question 2 Multiple Choice
A car tire is inflated to '35 psi.' A gauge reads the pressure inside as 35 psi. What is the approximate absolute pressure inside the tire?
A35 psia — gauge and absolute pressure are the same thing
B35 psia — atmospheric pressure is negligible and can be ignored
CApproximately 49.7 psia — gauge pressure is measured relative to atmosphere, so absolute pressure is gauge + atmospheric
DApproximately 20.3 psia — gauge pressure exceeds atmospheric, so absolute pressure is gauge minus atmospheric
Gauge pressure is measured relative to the local atmospheric pressure (approximately 14.7 psi at sea level). Absolute pressure is measured relative to a perfect vacuum. To convert: P_absolute = P_gauge + P_atm ≈ 35 + 14.7 ≈ 49.7 psia. When a gauge reads zero, the tire is at atmospheric pressure (not a vacuum). Option A confuses the two pressure references; option B ignores a 14.7 psi correction that is far from negligible. The distinction matters critically when applying the hydrostatic equation at boundaries where a fluid interfaces with the atmosphere.
Question 3 True / False
In a static fluid, the pressure at a given depth is greater directly below a heavy object resting on the fluid surface than at the same depth elsewhere in the fluid.
TTrue
FFalse
Answer: False
This is a violation of Pascal's law. In a static, connected fluid, pressure depends only on depth — not on what is above any particular column. A pressure change at any point is transmitted undiminished throughout the fluid. There is no 'shadow' of pressure below a heavy object floating or resting on the surface. The pressure at depth h is P₀ + ρgh everywhere at the same depth, where P₀ is the surface pressure. This uniform transmission is the principle behind hydraulic systems: a force applied to a small piston transmits equally to all other surfaces.
Question 4 True / False
Pressure in a static fluid is a scalar quantity, meaning it has magnitude but no directional component.
TTrue
FFalse
Answer: True
Pressure is indeed a scalar — it has a single numerical value at each point in the fluid, not a direction. This follows from the isotropy of pressure: at any given point in a static fluid, the pressure is the same in all directions. What has direction are the forces that pressure exerts on surfaces: pressure always acts perpendicular (normal) to whatever surface it contacts, but the pressure itself is directionless. Students sometimes confuse the force due to pressure (a vector, always normal to the surface) with pressure itself (a scalar). A fluid cannot sustain shear in static equilibrium, which is why no directional stress components exist.
Question 5 Short Answer
Why does pressure in a static fluid depend only on depth and fluid density, and not on the shape or total volume of the container?
Think about your answer, then reveal below.
Model answer: Pressure at any depth results from a force balance on a horizontal fluid element: the weight of the fluid column directly above it divided by its area. This ratio depends only on the height of that column and the fluid's density — not on how much fluid is off to the sides or what the container walls look like.
To derive P = P₀ + ρgh, consider a horizontal slab of fluid at depth h with area A. The forces acting on it are: pressure from above (P₀·A), pressure from below (P·A), and the weight of the slab itself (ρghA, downward). Static equilibrium requires these to balance: P·A = P₀·A + ρghA, giving P = P₀ + ρgh. Notice that the area A cancels — it doesn't matter whether A is large or small. Container walls exert horizontal forces on the fluid, but these horizontal forces cancel by symmetry and don't affect vertical pressure balance. The shape of the container changes where the walls push, but not the vertical force balance that determines pressure at depth.