1. Density and Pressure

General Definitions

• Density ($\rho$): Defined as mass per unit volume. It is an intrinsic property of a substance. $$ \rho = \frac{m}{V} $$

  • SI Unit: $kg/m^3$
  • Common conversion: $1 \text{ g/cm}^3 = 1000 \text{ kg/m}^3$ (Density of water $\rho_w$).

• Pressure ($P$): Defined as the magnitude of the normal force applied to a surface per unit area. $$ P = \frac{F}{A} $$

  • SI Unit: Pascal ($Pa$), where $1 Pa = 1 N/m^2$.

Pressure inside a Static Liquid

For a fluid at rest, pressure increases with depth due to the weight of the fluid column above. $$ P = P_0 + \rho g h $$

• $P$: Absolute pressure at depth $h$
• $P_0$: External pressure at the liquid surface (usually atmospheric pressure, $P_{atm}$)
• $\rho$: Density of the liquid (assumed constant)
• $g$: Acceleration due to gravity
• $h$: Depth below the surface (vertical distance)

💡 Common Applications

• Dams: Dams are built much thicker at the bottom than at the top. This is because water pressure increases linearly with depth ($P \propto h$), so the force exerted by the water is strongest at the base.
• Intravenous (IV) Fluids: IV bags must be placed above the patient's arm so that the liquid pressure is sufficient to overcome the patient's blood pressure and enter the vein.

Important Concept: Shape Independence (The Hydrostatic Paradox) The pressure at a given depth $h$ depends only on the vertical distance from the surface and the fluid density. It is independent of the shape of the vessel or the total amount of water in the vessel.

⚠️ Key Confusing Point: Students often think a wide tank creates more pressure at the bottom than a thin tube because it holds "more weight." This is incorrect. If the depth ($h$) is the same, the pressure is the same.

2. Pascal’s Principle and Hydraulics

Pascal's Principle

A change in pressure applied to an enclosed fluid is transmitted undiminished to every point of the fluid and to the walls of the container.

Hydraulic Lift

Since pressure is constant throughout a continuous, enclosed fluid at the same height: $$ P_{in} = P_{out} \Rightarrow \frac{F_1}{A_1} = \frac{F_2}{A_2} $$

• A small force $F_1$ applied to a small area $A_1$ can generate a large force $F_2$ on a large area $A_2$.

💡 Common Applications

• Car Brakes: When you push the brake pedal (small piston), the pressure is transmitted through the brake fluid to the brake pads (large pistons) at the wheels, multiplying the force to stop the car.
• Hydraulic Jacks: Used by mechanics to lift heavy cars with relatively little manual effort.

⚠️ Key Confusing Point: While force is multiplied, energy is conserved. The small piston must move a larger distance ($d_1$) to move the large piston a small distance ($d_2$). Work done is the same: $W = F_1 d_1 = F_2 d_2$.

3. Atmospheric Pressure

Atmospheric pressure is the weight of the air column above a specific point.

• Standard Atmosphere ($1 \text{ atm}$): $$ 1 \text{ atm} \approx 1.013 \times 10^5 \text{ Pa} \approx 101.3 \text{ kPa} $$

Mercury Barometer

A tube closed at one end is filled with mercury and inverted into a dish of mercury. The atmosphere pushes down on the open dish, supporting a column of mercury in the tube. $$ P_{atm} = \rho_{Hg} g h $$

• For standard atmospheric pressure, the height $h$ of the Mercury column is 760 mm (76 cm).

💡 Common Applications

• Drinking Straws: You don't actually "suck" liquid up. You lower the pressure inside your mouth/straw, and the atmospheric pressure pushing on the surface of the drink forces the liquid up into the straw.
• Suction Cups: Pressing a suction cup expels the air underneath. The higher atmospheric pressure outside pushes the cup firmly against the surface.