For more information, please see full course syllabus of AP Physics 1 & 2

# AP Physics 1 & 2 Continuity Equation for Fluids

III. Fluids: Lecture 3 | 7:00 min

This is a relatively short yet important topic. The continuity equation basically states that fluids travel at a constant rate, and are mass-conserving. A fluid will travel slower through a larger hole and faster through a smaller in order to conserve the volume of the fluid that’s displaced. If the fluid has uniform density, then mass is also conservatively displaced. This is much like Pascal’s Principle where it’ll be a very useful equation in determining aspects of one part of a system if you know another.

For more information, please see full course syllabus of AP Physics 1 & 2

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### Continuity Equation for Fluids

- The volume of fluid entering a full pipe must equal the volume of fluid leaving the pipe, even if the pipe's cross-sectional area changes. This is a restatement of the conservation of mass for fluids.

### Continuity Equation for Fluids

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Objectives 0:08
- Conservation of Mass for Fluid Flow 0:18
- Law of Conservation of Mass for Fluids
- Volume Flow Rate Remains Constant Throughout the Pipe
- Volume Flow Rate 0:59
- Quantified In Terms Of Volume Flow Rate
- Area of Pipe x Velocity of Fluid
- Must Be Constant Throughout Pipe
- Example 1: Tapered Pipe 1:44
- Example 2: Garden Hose 2:37
- Example 3: Oil Pipeline 4:49
- Example 4: Roots of Continuity Equation 6:16

### AP Physics 1 & 2 Exam Online Course

### Transcription: Continuity Equation for Fluids

*Hi everyone and welcome back to Educator.com.*0000

*Today we are going to continue our study of fluids as we talk about the continuity equation.*0003

*Our objectives are going to be to apply the continuity equation to fluids and motion.*0008

*To explain the continuity equation in terms of conservation of mass flow rate.*0012

*Conservation of mass for fluid flow. When fluids move through a full pipe, the volume of fluid entering the pipe must be equal to the volume of fluid leaving the pipe.*0017

*The Law of Conservation of Mass for Fluids.*0027

*This holds true even if the diameter of the pipe changes.*0030

*In short, what we call the volume flow rate remains constant throughout the pipe.*0034

*And we will look through a couple of applications of that here.*0055

*Volume flow rate. The volume of fluid moving through the pipe can be quantified in terms of volume flow rate.*0059

*The volume flow rate is the area of the pipe times the velocity of the fluid, and it must be constant throughout the pipe.*0065

*So over here on the left-hand side, if we are looking at a pipe with a changing diameter, we have Area1, where the fluid has Velocity1.*0072

*Over here on the right-hand side, we have Area2 and Velocity2.*0079

*A1V1, the volume flow rate on the left-hand side, must be equal to A2V2, the volume flow rate on the right-hand side.*0083

*What that means practically is that you must have a higher velocity or a faster flow over here and a slower flow over here.*0091

*Let us look at some examples and applications.*0104

*Water runs through a water main of cross-sectional area 0.4 square meters with a velocity of 6 meters per second.*0107

*Calculate the velocity of the water in the pipe when the pipe tapers down to a cross-secitonal area of 0.3 meters squared.*0114

*Well, continuity equation for fluid says A1V1 must equal A2V2.*0122

*Therefore, Velocity2 at the skinnier section of the pipe must be equal to A1 over A2 times V1.*0128

*Or 0.4 divided by 0.3 square meters times that 6 meters per second or 8 meters per second.*0129

*It gets a little narrower, it gets a little faster.*0150

*Let us take a look at the garden hose example.*0156

*A lot of folks have probably done this before.*0158

*As you are watering the garden or playing with the hose, you want the water to come out a little bit faster so you cover up the end of the nozzle with your thumb a little bit.*0160

*You decrease that cross-sectional area so that the water has to come out faster to maintain that volume flow rate.*0167

*In this problem, the water enters a typical garden hose of diameter 1.6 centimeters with the velocity of 3 meters per second.*0174

*Calculate the exit velocity of water from the garden hose when a nozzle of diameter half a centimeter is attached to the end.*0181

*First let us figure out what the cross-sectional areas are.*0188

*When it is entering the pipe, A1 is πr1 ^{2}, or π times. If our diameter is 1.6 centimeters, our radius must be 0.8 centimeters.*0192

*So that is 0.008 square meters, or an area of about 2.01 times 10 ^{-4} square meters.*0204

*Area 2 at the nozzle is πr2 ^{2} or π times. Well its diameter is 0.5 centimeters so its radius is half of that, 0.25 centimeters, 0.0025 meters squared.*0215

*Which is 1.96 times 10 ^{-5} square meters.*0231

*Now we can apply our continuity equation for fluids.*0239

*A1V1 equals A2V2. This implies then that V2 equals A1 over A2 times V1.*0245

*Or 2.01 times 10 ^{-4} square meters over 1.96 times 10^{-5} square meters.*0259

*All times V1 which was 3 meters per second, for a total of about 30.8 meters per second.*0270

*It comes out a lot faster when you decrease that area.*0282

*Let us take a look at an oil pipe line problem.*0287

*Oil flows through a pipe of radius (r) with speed (v).*0291

*Some distance down the pipe line, the pipe narrows to half its original radius.*0294

*What is the speed of the oil in the narrow region of the pipe?*0299

*Well, A1 we will call πR ^{2}. A2 is going to be πR/2^{2}, which is going to be πR^{2}/4 or π/4R^{2}.*0303

*Now as we apply the continuity equation for fluids, A1V1 = A2V2, which implies then that V2 = A1/A2(V1).*0325

*A1 = πR ^{2}, A2 = π/4R^{2} times V1 which we will just call V.*0341

*We are going to have some simplifications, R ^{2}, R^{2}, π, π.*0356

*We have 1/, 1/4 times V, which is going to be equal to 4V.*0361

*That is 4 times faster.*0370

*One last problem here.*0375

*So we look at the roots of the continuity equation, which statement below best describes the continuity equation for fluids?*0377

*Energy is conserved in a closed system? Mass is conserved in a closed system? Linear momentum is conserved in a closed system?*0384

*Angular momentum is conserved in a closed system? Or charge is conserved in a closed system?*0391

*Well, we are really talking about a mass conservation here.*0398

*The volume flow rate is basically saying, the continuity equation is saying that the mass that goes in must come out.*0404

*Therefore, mass is conserved in a closed system.*0409

*Hopefully this gets you a good start on the continuity equation for fluids.*0414

*Thanks for watching and make it a great day.*0418

1 answer

Last reply by: Professor Dan Fullerton

Sat Apr 6, 2013 10:33 AM

Post by Jawad Hassan on April 6, 2013

hey Dan,

just wanted to thank you, you are saving me in my current class, every thing is so simple and straight forward, my current teacher makes this way harder...