Problem 21 - College Physics

Question

If you dive to a depth of \(10 \mathrm{~m}\) below the surface of a lake, (a) what is the pressure due to the water alone? (b) What is the absolute pressure at that depth?

Step-by-Step Solution

Answer

(a) The pressure due to water is 98100 Pa. (b) The absolute pressure is 199425 Pa.

Step 1: Understanding Pressure due to Water

The pressure due to a fluid at a depth can be calculated using the formula: \( P = \rho \cdot g \cdot h \), where \( \rho \) is the density of the fluid (water in this case, taken as \( 1000 \text{ kg/m}^3\)), \( g \) is the acceleration due to gravity (\( 9.81 \text{ m/s}^2\)), and \( h \) is the depth (\( 10 \text{ m} \)).

Step 2: Calculating Pressure due to Water

Substitute the known values into the formula: \( P = 1000 \cdot 9.81 \cdot 10 = 98100 \text{ Pa} \). Thus, the pressure due to the water alone is \( 98100 \; \text{Pa} \) (Pascals).

Step 3: Understanding Absolute Pressure

Absolute pressure at a certain depth includes both the pressure due to the fluid and the atmospheric pressure above it. The atmospheric pressure at sea level is approximately \( 101325 \; \text{Pa} \).

Step 4: Calculating Absolute Pressure

Add the atmospheric pressure to the pressure due to the water: \( P_{\text{absolute}} = P_{\text{water}} + P_{\text{atmosphere}} = 98100 + 101325 = 199425 \; \text{Pa} \).

Step 5: Solution Verification

Verify the calculations to ensure all values are correctly substituted and added.

Key Concepts

Pressure Due to Fluid

Calculating the pressure exerted by a fluid, such as water, involves understanding how depth affects this pressure. The pressure at any given depth in a fluid can be determined by the formula:\[ P = \rho \cdot g \cdot h \]where:

• \( P \) is the pressure due to the fluid.
• \( \rho \) is the fluid density, which is typically \( 1000 \, \text{kg/m}^3 \) for water.
• \( g \) is the acceleration due to gravity, \( 9.81 \, \text{m/s}^2 \).
• \( h \) is the depth, such as \( 10 \, \text{m} \) underwater.

By substituting these values, the pressure due only to the water, at \( 10 \, \text{m} \) below water surface, becomes \( 98100 \, \text{Pa} \). This equation highlights how pressure increases proportionally with depth due to the weight of the water column above.

Absolute Pressure

Absolute pressure measures the total pressure that is exerted at a specific depth, including both the fluid pressure and the atmospheric pressure. In atmospheric contexts, it's crucial to consider not only the pressure from the fluid but also the air pressure that exists above it.
The absolute pressure is calculated by adding the atmospheric pressure to the pressure exerted by the fluid itself:\[ P_{\text{absolute}} = P_{\text{water}} + P_{\text{atmosphere}} \]In this situation:

• \( P_{\text{water}} = 98100 \, \text{Pa} \)
• \( P_{\text{atmosphere}} = 101325 \, \text{Pa} \)

The total absolute pressure at \( 10 \, \text{m} \) depth is thus \( 199425 \, \text{Pa} \). This calculation reflects combined forces acting at that point, both from under the surface and from the air above.

Density of Water

The density of a substance represents its mass per unit volume and is often expressed in kilograms per cubic meter (\( \text{kg/m}^3 \)).
For water, this constant density is typically taken to be \( 1000 \, \text{kg/m}^3 \).
This value is essential for calculations involving water pressure, as it directly affects how much pressure is exerted by a certain depth of water.
Density influences calculations in several ways:

• In formulas for pressure calculation, higher density means higher pressure at the same depth.
• A dense fluid will exert more weight on a submerged object, affecting buoyancy and pressure.

Understanding the density of water is critical when dealing with hydrodynamics and is one of the foundational blocks in calculating fluid pressures.

Atmospheric Pressure

Atmospheric pressure is the pressure exerted by the weight of air in the Earth's atmosphere.
At sea level, this pressure is approximately \( 101325 \, \text{Pa} \) (Pascals), but it can vary with altitude and weather conditions.
This pressure is always present and affects how we compute absolute pressure under water:

• It adds to the pressure exerted by the water itself.
• It's considered constant at a given height above sea level in calculations of submerged environments.

Atmospheric pressure is vital in understanding pressure differences and in designing systems like weather forecasting and altitude measurement. Realizing its ubiquity helps us grasp why it is added to fluid pressure to calculate total pressures experienced at depth.