Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

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 kg/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 Pa, what 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.

Short Answer

Expert verified
The minimum height \( h \) is approximately 0.580 meters.

Step by step solution

01

Understanding the Problem

We need to find the minimum height \( h \) such that the fluid, having a density of 1050 kg/m^3, moves from the reservoir into the vein, where the gauge pressure is 5980 Pa.
02

Applying the Hydrostatic Pressure Equation

Since the top of the reservoir is open to the air, the pressure at the top of the reservoir is equal to the atmospheric pressure. To move the fluid into the vein, the pressure due to the fluid column must at least equal the gauge pressure in the vein. The relationship is given by the hydrostatic pressure equation: \[ P = \rho g h \]Where \( P \) is the pressure due to the fluid column, \( \rho \) is the fluid density \( (1050 \text{ kg/m}^3) \), \( g \) is the acceleration due to gravity \( (9.81 \text{ m/s}^2) \), and \( h \) is the height.
03

Setting Up the Equation for Minimum Height

We set the hydrostatic pressure equal to the vein's gauge pressure to find the minimum height required for the fluid to enter the vein:\[ \rho g h = 5980 \text{ Pa} \]
04

Solving for Height

Rearrange the equation to solve for \( h \):\[ h = \frac{5980}{1050 \times 9.81} \]Calculate this value to determine \( h \).
05

Calculating the Result

Substitute the known values:\[ h = \frac{5980}{1050 \times 9.81} \approx 0.580 \text{ meters} \]Therefore, the minimum height \( h \) is approximately 0.580 meters.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Pressure calculation
Pressure calculation is a critical concept in hydrostatics. It involves understanding the force applied per unit area within a fluid. For fluids, the pressure often varies with depth because of the weight of the fluid above.

Some key points in pressure calculation include:
  • The formula to calculate pressure in a fluid is given by: \[ P = \rho g h \] Here, \( P \) denotes the pressure, \( \rho \) is the fluid density, \( g \) is the acceleration due to gravity, and \( h \) is the height or depth of the fluid column.
  • Pressure is fundamentally a reflection of how much force the fluid exerts on a surface due to gravity replacing the atmosphere at that point.
  • Since we're dealing with gauge pressure (pressure above atmospheric pressure), it's essential to recognize the interplay between atmospheric pressure and fluid pressure.
Understanding pressure calculation helps determine how deep or high a fluid needs to be to exert enough pressure to achieve a particular task, such as flowing into a vein as in the exercise example.
Fluid density
Fluid density is a measure of how much mass is contained within a given volume of fluid. It's a fundamental property that impacts how fluids behave and interact with other substances. For the exercise at hand:
  • Density is denoted by \( \rho \) and measured in \( \text{kg/m}^3 \). For the intravenous fluid, the density is given as 1050 \( \text{kg/m}^3 \).
  • Density directly influences pressure calculations since denser fluids exert more pressure at a given height compared to less dense fluids.
  • In practical applications, fluid density can influence the delivery and absorption rates in intravenous treatments or engineering designs in hydrostatic systems.
Therefore, understanding fluid density is crucial for predicting the pressure fluid can exert and ensuring it functions correctly in its application.
Gauge pressure
Gauge pressure is the pressure of a fluid relative to the ambient atmospheric pressure. It measures how much force the fluid in a system is exerting over what is naturally present in the environment.

In solving the exercise:
  • The gauge pressure inside the vein is given as 5980 Pa. This indicates the additional pressure the fluid needs to overcome to enter the vein.
  • The equation used in hydrostatics problems, like this one, often needs the gauge pressure because it directly affects how much pressure is needed beyond atmospheric pressure to achieve fluid flow.
  • Since the reservoir is open to air, the additional height of fluid needed is entirely to overcome this gauge pressure, confirming the calculations for \( h \).
By focusing on gauge pressure exclusively, we can tailor solutions that ensure safe and effective fluid delivery without unnecessary complications due to atmospheric variations.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Viscous blood is flowing through an artery partially clogged by cholesterol. A surgeon wants to remove enough of the cholesterol to double the flow rate of blood through this artery. If the original diameter of the artery is \(D\), what should be the new diameter (in terms of \(D\)) to accomplish this for the same pressure gradient?

A soft drink (mostly water) flows in a pipe at a beverage plant with a mass flow rate that would fill 220 0.355-L cans per minute. At point 2 in the pipe, the gauge pressure is 152 kPa and the cross-sectional area is 8.00 cm\(^2\). At point 1, 1.35 m above point 2, the cross-sectional area is 2.00 cm\(^2\). Find the (a) mass flow rate; (b) volume flow rate; (c) flow speeds at points 1 and 2; (d) gauge pressure at point 1.

Assume that crude oil from a supertanker has density 750 kg/m\(^3\). The tanker runs aground on a sandbar. To refloat the tanker, its oil cargo is pumped out into steel barrels, each of which has a mass of 15.0 kg when empty and holds 0.120 m\(^3\) of oil. You can ignore the volume occupied by the steel from which the barrel is made. (a) If a salvage worker accidentally drops a filled, sealed barrel overboard, will it float or sink in the seawater? (b) If the barrel floats, what fraction of its volume will be above the water surface? If it sinks, what minimum tension would have to be exerted by a rope to haul the barrel up from the ocean floor? (c) Repeat parts (a) and (b) if the density of the oil is 910 kg/m\(^3\) and the mass of each empty barrel is 32.0 kg.

Advertisements for a certain small car claim that it floats in water. (a) If the car's mass is 900 kg and its interior volume is 3.0 m\(^3\), what fraction of the car is immersed when it floats? Ignore the volume of steel and other materials. (b) Water gradually leaks in and displaces the air in the car. What fraction of the interior volume is filled with water when the car sinks?

You purchase a rectangular piece of metal that has dimensions 5.0 \(\times\) 15.0 \(\times\) 30.0 mm and mass 0.0158 kg. The seller tells you that the metal is gold. To check this, you compute the average density of the piece. What value do you get? Were you cheated?

See all solutions

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free