Pressure in Fluids

$\boxed{P = \frac{F}{A}}$

Where:

• $P$ = pressure (in pascals, Pa)
• $F$ = force (in newtons, N)
• $A$ = area (in square metres, m²)

1 pascal = 1 N/m² — one newton of force spread over one square metre.

Rearranging

$F = P \times A \qquad A = \frac{F}{P}$

Key Idea

• Same force, smaller area → higher pressure (e.g., stiletto heel vs flat shoe)
• Same force, larger area → lower pressure (e.g., snowshoes distribute weight)

The pressure at a depth in a liquid depends on the height of the liquid above, its density, and gravitational field strength:

$\boxed{P = h \times \rho \times g}$

Where:

• $P$ = pressure due to the liquid column (Pa)
• $h$ = height/depth of liquid (m)
• $\rho$ = density of the liquid (kg/m³)
• $g$ = gravitational field strength (N/kg)

Key Points

• Pressure increases with depth (deeper = more liquid above = more pressure)
• Pressure increases with density of the fluid (mercury exerts more pressure than water at the same depth)
• Pressure at the same depth acts equally in all directions
• The shape of the container does not affect the pressure at a given depth

Atmospheric pressure is caused by the weight of the air above us pressing down.

• At sea level: approximately 101,325 Pa (≈ 101 kPa or 1 atmosphere)
• Atmospheric pressure decreases with altitude (less air above)
• Acts in all directions (not just downwards)

Why Does It Exist?

Air has mass. The column of air above any point has weight, which creates pressure. The atmosphere extends about 100 km above Earth but most of the mass is in the lowest 10 km.

Demonstrating Atmospheric Pressure

• Collapsing can experiment: Heat a can with water inside, seal it, cool it. The steam condenses, reducing internal pressure. External atmospheric pressure crushes the can.
• Magdeburg hemispheres: Two hemispheres placed together with the air pumped out cannot be pulled apart because atmospheric pressure pushes them together.

Hydraulic systems use liquids to transmit forces. They work because liquids are virtually incompressible — pressure applied at one point is transmitted equally throughout the liquid.

How They Work

1. A small force is applied to a small piston (small area)
2. This creates a high pressure in the liquid ($P = F/A$)
3. The pressure is transmitted through the liquid to a large piston (large area)
4. The large piston experiences a large force ($F = P \times A$)

$\frac{F_1}{A_1} = \frac{F_2}{A_2}$

Force Multiplication

If the second piston has a larger area: $F_2 = F_1 \times \frac{A_2}{A_1}$

The output force is multiplied by the ratio of the areas.

Trade-off: The small piston must move a larger distance than the large piston. Work (energy) is conserved: the work done on the small piston equals the work done by the large piston.

$F_1 \times d_1 = F_2 \times d_2$

Applications

• Hydraulic brakes in cars
• Hydraulic jacks for lifting cars
• Excavators and heavy machinery
• Hydraulic presses

Upthrust is the upward force on an object submerged (partly or fully) in a fluid.

Upthrust exists because the pressure on the bottom of the object is greater than the pressure on the top (the bottom is deeper, so $P = h\rho g$ is greater).

$\text{Upthrust} = \text{weight of fluid displaced}$

This is Archimedes' Principle.

Floating and Sinking

• If upthrust = weight → the object floats
• If weight > upthrust → the object sinks
• An object floats when its average density is less than or equal to the fluid's density

• Pressure = force ÷ area: $P = F/A$ (Pa)
• Pressure in a liquid: $P = h\rho g$ — increases with depth and density
• Atmospheric pressure ≈ 101 kPa at sea level, decreases with altitude
• Hydraulic systems transmit pressure through incompressible liquids to multiply forces
• Upthrust = weight of displaced fluid; objects float when upthrust ≥ weight