All AP Physics 2 Resources
Example Questions
Example Question #1 : Flow Rate
An incompressible fluid flows through a pipe. At location 1 along the pipe, the volume flow rate is . At location 2 along the pipe, the area halves. What is the volume flow rate at location 2?
When the area halves, the velocity of the fluid will double. However, the volume flow rate (the product of these two quantities) will remain the same. In other words, the volume of water flowing through location 1 per second is the same as the volume of water flowing through location 2 per second.
Example Question #6 : Flow Rate
What is the volumetric flow rate of oil in a diameter pipe? The velocity of the oil is .
The volumetric flow rate of fluid is found using the equation:
Where is the velocity of the fluid and is the cross-sectional area of the space through which the fluid is flowing.
In this problem the cross-section of the pipe is a circle. The area of the cross-section is:
The volumetric flow rate is:
Example Question #13 : Fluid Dynamics
What is the volumetric flow rate of ethanol flowing through a square pipe with sidelength 4m? The velocity of the ethanol is .
The volumetric flow rate of fluid is found using the equation:
Where is the velocity of the fluid and is the cross-sectional area of the space through which the fluid is flowing.
In this problem the cross-section of the pipe is a square. The area of the cross-section is:
The volumetric flow rate is:
Example Question #3 : Flow Rate
A pipe narrows from a diameter to a diameter. What is the velocity of the fluid when it exits the pipe (at the end) if it entered the pipe at ?
The volumetric flow rate of fluid is found using the equation:
Where is the velocity of the fluid and is the cross-sectional area of the space through which the fluid is flowing. Use the continuity equation, we see that , therefore
In this problem, the cross-section of the pipe is a circle, which is
The area of the exit cross-section is:
Plug in these variables into the continuity equation and solve:
Example Question #4 : Flow Rate
Fluid enters a pipe at and exits the pipe at . What is the diameter of the pipe exit if the entrance has a diameter?
The volumetric flow rate of fluid is found using the equation:
Where is the velocity of the fluid and is the cross-sectional area of the space through which the fluid is flowing.
Use the continuity equation, we see that , therefore:
In this problem,
The cross-section of the pipe is a circle, which is:
Plugging our variables into the continuity equation gives us
Example Question #21 : Fluid Dynamics
Fluid flows through a pipe whose diameter goes from to . How are flow in the two parts of the pipe related?
The fluid in both sections of the pipe have the same velocity
The fluid in both sections of the pipe have the same volumetric flow rate
The fluid in both sections of the pipe are not related
The fluid in both sections of the pipe have the same cross-sectional area
The fluid in both sections of the pipe have the same volumetric flow rate
Based on the continuity equation, both parts of the flow must have the same volumetric flow rate.
Example Question #11 : Flow Rate
Molasses has a volumetric flow rate of . What volume of molasses has flowed after ?
The volumetric flow rate of fluid can be defined using the equation:
Where is the volume of the fluid and is the time the fluid is flowing. To solve this problem, time must be converted from minutes to seconds:
Using the volumetric flow rate equation we find that the volume of molasses is:
Example Question #11 : Flow Rate
Water is flowing through a diameter pipe at . Oil is flowing through a square pipe at . Which has the higher volumetric flow rate?
Water pipe
Both pipes have the same volumetric flow rate
Oil pipe
It is impossible to determine without knowing the density of the fluids
Water pipe
The volumetric flow rate of fluid is found using the equation:
Where is the velocity of the fluid and is the cross-sectional area of the space through which the fluid is flowing. In this problem the cross-section of the water pipe is a circle. The area of the cross-section is:
The volumetric flow rate is:
The cross-section of the oil pipe is a square. The area of the cross-section is:
The volumetric flow rate is:
The water pipe has the larger volumetric flow rate.
Example Question #21 : Fluid Dynamics
Consider the case in which a fluid is flowing through a cylindrical tube. Which of the following would be expected to not increase the volume flow rate?
Increasing the length of the pipe
Increasing the pressure difference on either side of the pipe
Decreasing the viscosity of the fluid in the pipe
Increasing the radius of the pipe
Increasing the ratio of laminar to turbulent flow
Increasing the length of the pipe
For this question, we need to consider the flow of a fluid through a cylindrical pipe. We then need to determine which parameter, when changed, will decrease the flow rate of fluid through the pipe.
First, let's consider a few variables that are implicated in fluid flow. In fact, the flow of a fluid through a pipe (or a connection of pipes) is analogous to a circuit and Ohm's law. For example, the expression for Ohm's law is as follows:
From this expression, we can see that the voltage difference (driving force for movement of charge) is equal to the current (flow of charge) multiplied by the resistance (which impedes the flow of charge). Similarly, the flow of a fluid through a pipe can be expression as follows:
Where is equal to the pressure difference, is equal to flow rate, and is equal to resistance. This expression takes the same form as Ohm's law. The pressure (driving force for movement of the fluid) is equal to the flow rate (movement of fluid) multiplied by the resistance (which, again, impedes the flow of the fluid).
Making this comparison between Ohm's law and flow rate should help make it easier to remember. But to answer the question, let's rearrange the expression slightly.
In this form, we can see that as the pressure difference is increased, so too does the flow rate increase. Moreover, flow rate is higher when resistance is lower. We need to keep these things in mind when determining what will reduce flow rate. Since increasing the pressure difference increases the flow rate, we can rule this answer choice out.
Next, let's see what things contribute to resistance. While the equation for resistance is quite complicated, we can remember a few generalities. First, the resistance of flow is inversely related to the radius of the pipe. In other words, as the radius increases, there is more room for the fluid to flow through, and thus the flow rate increases.
When the length of the tube increases, the resistance also increases. Think of it this way: it's easier to move a given amount of water through a straw than it is a garden hose. A big reason for this is that the increased length of the tube provides more area to come into contact with the moving fluid, which thus allows more opportunity for friction between the fluid and the walls of the pipe.
Fluids flow fastest when their flow is laminar. Turbulent flow is characterized as disordered and involves "wasting" energy through the movement of liquid molecules in directions other than the main direction of flow, which also impedes other fluid molecules and causes them to bump into the walls of the container/other fluid molecules irregularly, ultimately disrupting flow.
Finally, increasing the viscosity of the fluid also increases its resistance, and thus decreases its flow rate. This is because viscosity is a measure of the frictional interactions between molecules of the fluid itself. Think of it this way: it's harder to get something like syrup (very viscous) to flow as opposed to water (much less viscous). Thus, when we decrease the viscosity of a fluid, the resistance to flow decreases, and flow rate increases.
Example Question #11 : Flow Rate
Water is flowing through a circular tube with cross sectional area at a rate of . The cross section of the tube gradually decreases to as the tube gains in height. What is the new velocity of the water?
Assume
We will solve this problem in two steps. First, will use the law of continuity:
Rearranging for velocity 2:
Now we can use the expression for conservation of energy to adjust this velocity for the change in height:
Assuming an initial height of 0, we get:
Plugging in expressions, we get:
Rearranging for final velocity:
Plugging in our values, we get:
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