Problem 11 - University Physics with Modern Physics

Question

In intravenous feeding, a needle is inserted in a vein in the patient's arm and a tube leads from the needle to a reservoir of fluid (density 1050 \(\mathrm{kg} / \mathrm{m}^{3} )\) located at height \(h\) above the arm. The top of the reservoir is open to the air. If the gauge pressure inside the vein is 5980 \(\mathrm{Pat}\) is the minimum value of \(h\) that allows fluid to enter the vein? Assume the needle diameter is large enough that you can ignore the viscosity (see Section 12.6\()\) of the fluid.

Step-by-Step Solution

Answer

The minimum height \(h\) is approximately 0.58 meters.

Step 1: Understand the Problem

We need to find the minimum height \(h\) of the fluid reservoir that will allow fluid to enter the vein. This problem is about hydrostatic pressure, where the fluid at a certain height will exert pressure due to gravity. We use gauge pressure here, and atmospheric pressure cancels out because it's the same at both the reservoir and the vein top.

Step 2: Apply Hydrostatic Pressure Formula

The hydrostatic pressure exerted by a fluid column can be calculated using the formula: \( P = \rho gh \), where \( P \) is the pressure exerted by the fluid column, \( \rho \) is the density of the fluid, \( g \) is the acceleration due to gravity (approximately \(9.81 \/ \mathrm{m\/s^2}\)), and \( h \) is the height of the fluid column.

Step 3: Set up the Equation

Given that the minimum pressure \(P\) inside the vein is 5980 Pa, we set up the equation using the hydrostatic pressure formula: \[ P = \rho g h \] Substitute given values: \[ 5980 = 1050 \times 9.81 \times h \]

Step 4: Solve for Height \(h\)

Rearrange the equation to solve for \(h\): \[ h = \frac{5980}{1050 \times 9.81} \] Calculate the right-hand side to find the minimum height.

Step 5: Calculate Height \(h\)

Perform the calculation: \[ h = \frac{5980}{1050 \times 9.81} \approx 0.58 \, \mathrm{m} \] Thus, the minimum height \(h\) is approximately 0.58 meters.

Key Concepts

Fluid Density

Fluid density is an important concept when dealing with hydrostatics and intravenous feeding. Density is defined as mass per unit volume of a substance. It is typically represented by the Greek letter \( \rho \). In our scenario, the fluid density is given as 1050 \( \text{kg/m}^3 \). This implies how much mass there is in one cubic meter of the fluid.
Understanding fluid density helps us comprehend how different fluids will behave under the same conditions. High density means more mass and potentially more pressure exerted at a certain depth or height. In medical scenarios such as intravenous feeding, fluid density affects how easily the fluid can be administered through the tube into a patient's system.

• High density - More mass in the same volume
• Affects pressure within the system
• Relevant for calculating hydrostatic pressure

Gauge Pressure

Gauge pressure refers to the pressure in a system, excluding atmospheric pressure. It is measured relative to atmospheric pressure, which is why it can be negative if the measured pressure is below atmospheric pressure. In the context of intravenous feeding, gauge pressure is used to determine how much additional pressure the fluid reservoir must exert to overcome the pressure within the vein.
In our problem, the gauge pressure inside the vein is given as 5980 Pa. This means that any external setup must exert this much pressure to successfully push the fluid into the vein. Gauge pressure is essential in medical settings as it ensures that fluids are delivered at the right pressure, avoiding injury or discomfort.

• Pressure measured relative to atmosphere
• Important for medical applications
• Ensures accurate fluid delivery

The difference between gauge and absolute pressure must always be considered for precise fluid regulation.

Intravenous Feeding

Intravenous feeding is the process of delivering nutrients directly into the bloodstream through a vein. It is a common medical procedure used when a person cannot eat or drink normally. A needle and tube system delivers the fluid from a reservoir into the patient's bloodstream.
In the provided problem, determining the correct reservoir height is critical to ensuring the fluid enters the vein properly. The system needs to overcome the vein's internal pressure due to blood.

• Direct fluid delivery
• Overcomes physiological pressures
• Critical for patient health and nutrition

Understanding the principles of hydrostatics ensures successful and safe intravenous feeding.

Hydrostatic Equation

The hydrostatic equation is fundamental in solving problems like intravenous feeding. This equation calculates the pressure exerted by a fluid column based on height and density.
The formula is \( P = \rho gh \), where \( P \) is the pressure, \( \rho \) is the fluid density, \( g \) is the acceleration due to gravity, and \( h \) is the height of the fluid column. In our exercise, this equation helps find the minimum height \( h \) required for the fluid in the reservoir to exert enough pressure to enter the vein.

• Key formula in fluid mechanics
• Relates density, gravity, and height to pressure
• Used to determine force exerted by fluids

Knowing how to apply the hydrostatic equation ensures appropriate design of medical equipment and procedures.