8.2 Variation of Pressure with Depth

• Pressure in fluids increases with depth due to the weight of the fluid above.


  True
• What is the value of acceleration due to gravity used in the pressure formula?

  9.8 \, \text{m/s}^2</latex>
• The density of a fluid affects the pressure at a given depth.


  True
• In the pressure formula, \( \rho \) represents the density
• What are the units for pressure in the SI system?

  Pascals (Pa)
• What is the pressure at a depth of 15 meters in seawater, given \( P_0 = 101,325 \, \text{Pa} \) and \( \rho = 1025 \, \text{kg/m}^3 \)?

  $251,712.5 \, \text{Pa}$
• What is the pressure at a depth of 15 meters in seawater, given \( P_0 = 101,325 \, \text{Pa} \) and \( \rho = 1025 \, \text{kg/m}^3 \)?

  251,712.5 Pa
• To apply the equation \( P = P_0 + \rho gh \), we'll solve two example problems
• The initial pressure calculated at a depth of 10 meters in water is 102,000 Pa.


  True
• Pressure is the force applied per unit area
• What is the relationship between pressure and depth in fluids?

  $P =$$P_{0} +$$\rho gh$
• What is the value of acceleration due to gravity used in the pressure variation formula?

  $9.8 \, \text{m / s}^{2}$
• Steps in deriving the equation for pressure variation with depth

  1️⃣ Consider a small cube of fluid at depth \( h \).

  2️⃣ Calculate the pressure using \( P = \frac{F}{A} \).

  3️⃣ Find the force exerted by the fluid above: \( F = \rho A h g \).

  4️⃣ Simplify the pressure formula to \( P = \rho gh \).

  5️⃣ Add the initial pressure \( P_0 \) to get $P =$$P_{0} +$$\rho gh$.
• The weight of the fluid above a small cube exerts pressure on it.

  True
• What is the formula for the force exerted by the weight of the fluid above?

  $F =$$\rho A h g$
• The initial pressure at the surface is added to the pressure calculated at depth.


  True
• What is the formula for pressure at depth in a fluid?

  $P =$$P_{0} +$$\rho gh$
• In the formula for pressure at depth, \( P_0 \) represents the initial pressure
• What does \( h \) represent in the pressure formula?

  Depth in meters
• Match the type of pressure with its example:

  Tire Pressure ↔️ Keeps car wheels inflated

  Atmospheric Pressure ↔️ Affects weather patterns
• Steps to derive the equation for pressure variation with depth

  1️⃣ Consider a small cube of fluid at depth \( h \).

  2️⃣ Calculate pressure at depth \( h \) using \( P = \frac{F}{A} \).

  3️⃣ Determine the force exerted by the weight of the fluid above: \( F = \rho A h g \).

  4️⃣ Derive the pressure as \( P = \rho gh \).

  5️⃣ Add the initial pressure \( P_0 \) to get \( P = P_0 + \rho gh \).
• The formula for pressure variation with depth assumes uniform fluid density.


  True
• If the pressure at a depth of 10 meters in water is \( 200,000 \, \text{Pa} \) and \( \rho = 1000 \, \text{kg/m}^3 \), the initial pressure is 102,000 Pa.
• Calculate the initial pressure if the pressure at a depth of 10 meters in water is \( 200,000 \, \text{Pa} \) and \( \rho = 1000 \, \text{kg/m}^3 \).

  102,000 Pa
• The pressure at a depth of 15 meters in seawater is 251,712.5 Pa.


  True
• What is the formula for pressure in terms of force and area?

  $P =$$\frac{F}{A}$
• Pressure increases with depth in a fluid.


  True
• Match the variable with its description:

  \( P \) ↔️ Pressure at depth

  \( P_0 \) ↔️ Initial pressure

  \( \rho \) ↔️ Fluid density

  \( g \) ↔️ Acceleration due to gravity
• To derive the equation for pressure variation with depth, consider a small cube of fluid
• The equation for pressure variation with depth in a fluid is derived using the concept of a small cube
• The pressure at a depth \( h \) can be calculated using \( P = \frac{F}{A} \), where \( F \) is the force
• The formula for pressure at depth \( h \) in a fluid is \( P = \rho gh \), where \( \rho \) is the density
• What is the complete formula for pressure at depth including initial pressure \( P_0 \)?

  $P =$$P_{0} +$$\rho gh$
• The pressure at a depth of 15 meters in seawater with \( P_0 = 101,325 \, \text{Pa} \) and \( \rho = 1025 \, \text{kg/m}^3 \) is 251,712.5
• What is the initial pressure if the pressure at a depth of 10 meters in water is \( 200,000 \, \text{Pa} \) and \( \rho = 1000 \, \text{kg/m}^3 \)?

  102,000 Pa
• What is the mathematical expression for Pascal's Principle?

  $P =$$\frac{F_{1}}{A_{1}} =$$\frac{F_{2}}{A_{2}}$
• One key aspect of Pascal's Principle is uniform pressure distribution in a closed fluid system
• What is the pressure increase at a depth of 10 meters in water, given a density of 1000 kg/m³?

  98,000 Pa
• The pressure increase at a depth of 10 meters in air, given a density of 1.225 kg/m³, is 120 Pa.
• Match the variable with its description:

  \( P \) ↔️ Pressure at depth

  \( P_0 \) ↔️ Initial pressure

  \( \rho \) ↔️ Fluid density

  \( g \) ↔️ Acceleration due to gravity

  \( h \) ↔️ Depth