Practicals Manual - U of T Physics
[pic] |PHY131H1S
Practicals Manual |[pic] | |
|Department of Physics | |University of Toronto |
| |January to April, 2009 | |
Welcome to the Physics Practicals! We have devised a number of Activities and Projects which will help you to learn a lot of Physics. They will also help you to do well on the tests and exam of the course. We are very excited about this new way of helping you to learn Physics, and hope you find your time in the Practicals to be fun and productive.
The course web-site has the most up-to-date contact information, handouts, schedules and information:
The course coordinator and lecturer is
Jason Harlow, Office: MP129-A, Phone 416-946-4071.
The materials in this book were mainly developed by
David Harrison, Office: MP121-B, Phone 416-978-2977
The co-coordinator of the course is
Pierre Savaria, Office: MP129-E, Phone 416-978-4135.
The course administrator is April Seeley, Office: MP129-E, Phone 416-946-0531.
Email addresses are listed on the course web-site and on the Physics Department directory at .
Course staff will endeavour to respond to email inquiries from students within 2 days. If you do not receive a reply within this period, please resubmit your question(s) and/or phone (leave a message if necessary). Please note that some servers (such as hotmail) can be unreliable in both sending and receiving messages.
|Table of Contents |
|Section |Page |
|Introduction |1 |
|Mechanics Module 1 |7 |
|Scientific Method Module |15 |
|Mechanics Module 2 |17 |
|Mechanics Module 3 |28 |
|Mechanics Module 4 |37 |
|Teamwork Module |42 |
|Mechanics Module 5 |50 |
|Mechanics Module 6 |59 |
|Oscillations Module |69 |
|Numerical Approximation Module |85 |
|Fluids Module |94 |
Schedule (preliminary)
Students will meet once per week in MP126-P on Thursdays or Fridays for two hours.
|Practical |Dates |Topics, Activities |
|Session # | |Schedule is Subject to Change |
| |Jan 8, 9 |NO PRACTICALS THIS WEEK |
|1 |Jan 15, 16 |Introduction |
| | |Mechanics Module 1 |
|2 |Jan 22, 23 |Error Analysis Assignment due |
| | |Scientific Method Module |
| | |Mechanics Module 1 continued |
| | |Mechanics Module 2 |
|3 |Jan 29, 30 |Written Homework #1 due |
| | |Mechanics Modules 2 continued |
|4 |Feb 5, 6 |Mechanics Module 3 |
| |Feb 12, 13 |NO PRACTICALS THIS WEEK – Extra office hours for test prep. |
| |Feb 19, 20 |Reading week. Then test on Feb.24 evening. |
|5 |Feb 26, 27 |Scrambling teams |
| | |Teamwork module |
| | |Numerical Approximation Module |
|6 |Mar 5, 6 |Mechanics Module 4 |
|7 |Mar 12, 13 |Mechanics Module 5 |
|8 |Mar 19, 20 |Written Homework #2 due |
| | |Mechanics Module 6 |
|9 |Mar 26, 27 |Oscillations Module |
|10 |Apr 2, 3 |Fluids Module |
The Error Analysis Assignment is a series of online tutorials which must be viewed with a computer. There is a link from the Practicals tab of the main course web page. PHY131 should please write the answers on a paper print-out of the 3-page “Answer Sheet”, which is linked as a PDF on the first page of the assignment. The answers, done by individual students, are due in the 2nd practicals session.
How the Practicals Work
You will be meeting for 2 hours every week in room MP126P, which is in the back of MP126. Each Group will have a maximum of 36 students. You will be working in a Team with up to three of your classmates. There will be two Teaching Assistant Instructors present for each Practical.
Your Team will keep a single lab book, which is to be a complete record of everything you did, what you and your teammates thought it meant, and what conclusions you have drawn from your work.
Each Practical session will include time for student questions and discussion. However the “heart” of the Practicals will be a series of Activities.
Every week you will be doing Activities based on the material currently being discussed in class. Often the Activities will be based on material that has already been discussed in class, but sometimes the Activities may be used to introduce material that has not yet been talking about in class. In addition, you will be doing three “value added” Modules, that we believe are important for your overall learning about science in general and Physics in particular. These are:
• A Module on the Scientific Method
• A Module on effective Teamwork
• A Module on Numerical Approximation
For each Practical session two members of each Team will serve the following roles:
Facilitator. This person, a different individual each week, is responsible for keeping the Team on track with the Activities. When the entire Practical group discusses some topic, the Facilitator will be the Team’s primary spokesperson.
Recorder. This person, also a different individual each week, takes primary responsibility for recording all work, speculations, conclusions etc. in the lab notebook.
Evaluation and Marks
The Practicals will count for 20% of your mark in PHY131.
All marks will be given on an integer scale from 0 to 4:
0. Missing work.
1. Seriously deficient.
2. Requires improvement.
3. The standard mark indicating good work
4. Exceptional. We will be very stingy in awarding marks of 4.
Each mark component has a weight, and the mark times the weight will be added to generate a Practical mark. The total number of weights of all components is 20. The one exception to this marking system is the Error Analysis Assignment. It is marked out of 100.
Attendance at the Practical is vital for your learning. We will deduct the cube of the number of un-excused absences from the final Practical mark.
Here are the components and their weights:
1. Notebook Mark 1 (0 Weights). After the first Practical the lab books will be collected and marked. However, this mark will not count towards your Practical mark. Instead it is intended to make our standards and requirements clear to you.
2. Error Analysis Assignment (1 Weight). You will do this assignment individually. It is due at Practical Session #2.
3. Scientific Method Module (1 Weight). This Module will be done during the second practical, and will be marked.
4. Written Homework #1 (2.5 Weights). You will do this assignment outside of class in collaboration with the members of your first Practicals team. It is due at Practical Session #3.
5. Notebook Mark 2 (6 Weights). After the last Practical before Test, a selection of Activities from Practical sessions completed so far will be chosen to be marked. The decision of which Activities will be marked will be chosen more-or-less randomly after the books have been collected. All Teams will have the same Activities marked.
6. Numerical Approximation Module (1 Weight).
7. Written Homework #2 (2.5 Weights). You will do this assignment outside of class in collaboration with the members of your second Practicals team. It is due at Practical Session #8.
8. Notebook Mark 3 (6 Weights). At the end of the term a selection of the Mechanics, Oscillations and Fluids Activities you have done since the Test will be chosen to be marked. The decision of which Activities will be marked will be chosen more or less randomly after the books have been collected. All Teams will have the same Activities marked.
Computers and Networks
The Practical server is: feynman.physics.utoronto.ca. You will access the server using your UTORid and password. You will have access to three folders on this server:
Your home directory. You have read and write privileges for this directory.
Your team directory. All members of your team have read and write privileges here.
public. This is an area of the server containing documents, computer programs, etc. Everyone has read privileges for this directory.
Note: you should never save work on the local PC. These discs will be ruthlessly purged on a regular basis.
Remote Access
You may access the server at: . You may upload and download files from your computer to the server.
Printing
There is a colour printer in the Practical Room. You may choose to print either in colour or black and white by choosing the appropriate printer in the print dialog. We charge for printing using your TCard. We charge:
10 cents per page for black and white printing.
15 cents per page of colour printing.
We do not (yet) have facilities in the building to add dollar values to your card. The locations of cash-to-card locations is at:
At present the nearest location is the Main Floor of the Earth Science building, just across Huron Street.
Datasets
All datasets in the Practicals have a standard uniform format. This section describes that format.
The dataset file is text.
1. The first line of the file is the title of the dataset.
2. The second line of the file names the variables of the data. The names are separated by tabs. In the examples below we represent a tab with:
3. The third and subsequent lines of the file contain the data. Each datapoint is on a separate line and the values are separated by tabs.
Thus, the dataset can be edited with a text editor or a spreadsheet program such as Excel.
There are four cases for the number of variables in the dataset.
One Variable
If only one value is given for each datapoint, it is the dependent (i.e. y) variable. In this case the values of the independent (x) variable are assumed to be 1, 2, 3, … in order.
Here is an example of such a dataset:
|Balonium decay values |
|Counts per second |
|50 |
|32 |
|27 |
|15 |
|11 |
|8 |
Two Variables
In this case the first column contains the values for the independent (x) variable and the second column the values for the dependent (y) variables. For example:
|Student collected data on pressure-temperature values |
|Pressure (cm Hg)Temperature (C) |
|65-10 |
|7517 |
|8642 |
In the above denotes the TAB character.
Three Variables
If there are three variables, the third one is the error in the dependent (y) variable.
|Thermocouple calibration data |
|Temp (C)Voltage (Volts)errV |
|0-0.890.05 |
|5-0.690.05 |
|10-0.530.05 |
Four Variables
Now there are explicit errors in both coordinates of the data. The first column contains the name and values of the independent (x) variable, the second column contains the name and values of the error in the independent variable, the third column contains the name and values of the dependent (y) variable and the fourth column the name and values of the error in the dependent variable.
|Pearson’s Data with York’s Weights |
|XerrXYerrY |
|00.03165.91 |
|0.90.03165.40.746 |
|1.80.04474.40.5 |
Mechanics Module 1
Student Guide
Concepts of this Module
• Scaling
• Dimensions
• Fermi Problems
• Introduction to Experimental Uncertainties
• Kinematics in One Dimension
• Motion Diagrams
• Setting up Newtonian Dynamics
The Activities
Activity 1
A sculptor is making a statue of a duck. She first creates a model. To make the model requires exactly 2 kg of bronze. The final statue will be 5 times the size of the model in all three dimensions. How much bronze, in kg, will she require to cast the final statue?
You may find it helpful to think about the model being constructed of Lego blocks, with the final statue made of Lego blocks that are 5 times the size in each dimension as the ones used to make the model.
Activity 2
When the sculptor finished making her model of the duck statue, she gave it 2 coats of varnish. This took exactly one can of varnish. How many cans of varnish will she need to give the final statue 2 coats of varnish?
Activity 3
Surprisingly, the units of all physical quantities can be defined in terms of combinations of only four fundamental units: a unit for length, mass, time, and electric current. In the SI system the units are:
• Second, s: the time required for 9,192,631,770 oscillations of the radio wave absorbed by the cesium-133 atom.
• Meter, m: the distance traveled by light in a vacuum in 1/299,792,458 of a second.
• Kilogram, kg: the mass of the international standard kilogram, a polished platinum-iridium cylinder stored in Paris.
• Ampere, A: the constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 m apart in vacuum, would produce between these conductors a force equal to
2 x 10-7 newton per meter of length.
Mort the politician has a not-so bright idea that we could save money by simplifying the standards for units. Instead of having a unit of length be fundamental, the politician suggests having a unit of volume as fundamental. Of course this unit of volume would be called a mort. Then, instead of a difficult to measure and expensive separate standard for length we could define the volume of the standard kilogram to be exactly 1 mort.
In this system of units, what is the unit of density?
What is the density of the standard kilogram in kg/mort?
The density of the standard kilogram is about 21,500 kg/m3. The density of water is 1000 kg/m3. What is the density of water in kg/mort?
In this system of units length is now a derived quantity. What is its relation to the mort?
You have a replica of the standard kilogram and an object of unknown material with a similar volume. How might you actually measure the volume of this object to determine if its volume is greater than, less than, or equal to one mort?
Activity 4
The ancient Greeks built a temple to Apollo on the island of Delos. It was 11 m wide, 24 m long, and 10 m high. In 427 B.C. a plague ravaged Athens, and the Athenians consulted the oracle on Delos, who demanded that they double the size of the temple.
a) What is the original volume of the temple?
b) The Athenians re-built the temple by doubling the size of each dimension of the temple. What was the volume of the new temple?
c) The Athenians consulted to oracle again, who said “You have not doubled the size of god’s temple, as he demanded of you.” What mistake did the Athenians make?
d) What would be the dimensions of the temple that the oracle wanted the Greeks to build?
Activity 5
How many musical notes are played on an average radio station in a given year?
Activity 6
A useful visualization technique in studying motion is called a motion diagram. We will be using these diagrams frequently in this course.
For example, consider an apple that is dropped from rest at some height above the ground.
For many objects in translational motion we can ignore the details of the object itself and model the object as an ideal particle and draw it as a simple dot. We number each dot to show the order in which the apple was at the positions indicated. The same amount of time elapses between each dot and the next one. The figure to the right shows the motion diagram for the apple in free fall.
Four motion diagrams are shown below. One is of a car moving to the right at constant speed, one is of a car moving to the left at constant speed, one is a car accelerating to the right away from a stop light that has just turned green, and one is a car moving to the right and slowing down as it approaches a stop sign. Which motion diagram corresponds to which case?
[pic]
Activity 7
If a motion diagram represents the position of an object every second, then the distance between each dot and the next is numerically equal to the average speed of the object during that one second interval. For the motion diagrams of Activity 6, draw a line from each dot to the next representing the magnitudes of these speeds. Put an arrowhead on each line indicating the direction of the motion.
Imagine that two of the dots in the motion diagram are separated by 0.15m. If the second dot is the position of the object 1.0 second after the position of dot 1, what is the average speed of the object during this one second interval?
Imagine that the two dots of Part B, 0.15 m apart, represent the positions of the objects for a time interval of 0.50 seconds. Now what is the average speed of the object during the half-second interval?
In Part A you “connected to dots” of the motion diagram. If the motion diagram represents the position of the object every millisecond, what is the relationship between the length of the line from each dot to the next and the average speed of the object during that millisecond?
Activity 8
Here are some made up data for the x component of the position of an object at various times:
|Time (s) |Position (m) |
|0.00 |0.002 |
|0.10 |0.111 |
|0.20 |0.385 |
|0.30 |0.892 |
|0.40 |1.613 |
|0.50 |2.501 |
|0.60 |3.612 |
Sketch a graph of position vs. time. . Make the horizontal axis the time and the vertical axis the position.
Is it reasonable to “connect the dots” with a smooth line in the graph you sketched? If yes, what assumption is being made about the motion of the object? If no, why?
Sketch a motion diagram of the motion of the object.
Calculate the displacements of the object for each 0.1 s interval.
How does the number of displacements you calculated compare to the number to the number of data points in the position-time data?
From Part D, calculate the x components of the average velocities of the object for each 0.1 s interval.
Consider the first of the average velocity values from Part F. At what time does the object have this value of the average speed? Is the value of the time a single value or a range of values? Why?
Sketch a graph of the average velocity versus time. Make the time the horizontal axis.
From your result from Part F calculate the x component of the average acceleration of the object for each 0.1 s interval. How does the number of calculated values of the average acceleration compare t the number of data points in the position-time data?
Sketch a graph of the average acceleration versus time. Make the time the horizontal axis.
What does the data indicate about the acceleration of the object?
Activity 9
Imagine that the data from Activity 8 were taken with a computerized data acquisition system. The system has nearly perfect accuracy, but the precision of each distance measurement is ± 0.020 m. What are the corresponding uncertainties in the calculated values of the displacements, velocities, and accelerations/
Activity 10
An experiment to determine whether Energizer or Duracell batteries last longer could measure the number of hours two AA batteries from each brand will run a tape player. Here is some made up data:
| |Trial 1 |
|0.01000 |1.00001 |
|0.10000 |1.00063 |
|π/4 |1.03997 |
|π/2 |1.18034 |
|2.000000 |1.32890 |
|3.000000 |2.57123 |
|π |∞ |
There are a few ways to avoid becoming mired in all the mathematics:
1. Use numerical approximation.
2. Build and measure a real pendulum.
3. Do even more mathematics to simplify Eqn (6) into an approximate form that can be solved with a simple calculator.
In Activity 9 we will use numerical approximation, and Activity 10 we will use a real pendulum. If you wish to see the sort of gyrations necessary to get a formula that can be solved with a calculator, see for example F.M.S. Lima and P. Arun, “An accurate formula for the period of a simple pendulum oscillating beyond the small angle regime”, American Journal of Physics 74(10), 892 – 895 (2006), .
Activity 9
Here are some values of the period of a simple pendulum for a length L = 1.0000 m taking g = 9.8000 m/s2 using Eqn. 5 and the values of Table 1.
Table 2
|θmax (rads) |T (s) |
|0.01000 |2.00710 |
|0.10000 |2.00835 |
|π/4 |2.08732 |
|π/2 |2.36905 |
|2.000000 |2.66723 |
|3.000000 |5.16070 |
|π |∞ |
A listing of the code for Pendulum.py appears in Appendix 1. We assume that the mass is connected to the support by a massless rigid rod. The algorithm used to approximate this system is very similar to the one used for the spring-mass system investigated in the Numerical Approximation Module. Examine the code and describe in your own words how the algorithm approximates the motion of the pendulum.
The code also estimates the period of the oscillation. Describe in your own words how the estimation is done.
The “master” copy of Pendulum.py is located at:
Feynman:Public/Modules/Oscillations
Copy the file to your Team’s area on the server. Start the IDLE for VPython program and open the copy of Pendulum.py. Use Run / Run Module or press the F5 key on your keyboard to start the animation. How does the estimated value of the period compare to the value in Table 1? Can you think of a better way to determine the period of the simulation?
Modify the code so that when the animation starts it prints the maximum amplitude and the timestep of the approximation. Test your changes to make sure that they work. You will wish to know that when you change the code and run the animation VPython will over-write the file with the new version.
A tip: when you start the animation, two windows are opened, a Python Shell which shows the results of all print statements in the code and a window of the animation. To stop the animation close the animation window, but leave the Python Shell window open. This will be useful for Part E.
Does the numerical approximation agree with the values of Table 2? How does the approximation do for different values of the timestep? You may wish to print the window of the data for the estimated periods for various maximum amplitudes and timesteps and staple it into your lab book.
Activity 10
For a real pendulum, things are not quite so simple. In the figure we show a real pendulum. The distance between the pivot point and the centre of mass is L, the moment of inertia of the object about the pivot is I, and the total mass of the object is m. Often this is called a physical pendulum.
Using the usual convention that positive torques cause counter-clockwise rotations, the torque exerted on the pendulum is:
[pic] (7)
Newton’s 2nd Law for rotational motion is:
[pic] (8)
Thus, the equation of motion is:
[pic] (9)
Note that except for differences in the symbols this has the same mathematical structure as Eqn (1) for a simple pendulum. Therefore within symbol changes the solution is also the same. In particular, in the small angle approximation the motion is simple harmonic and the period of oscillation is:
[pic] (10)
Similarly, for oscillations where the maximum amplitude is not small Eqn (6) is also true:
[pic] (11)
Therefore the ratios of T/T0 from Table 1 are also true for this case.
Here you will side-step trying to solve Eqn (11) by taking data on how the period of a real physical pendulum varies with its maximum amplitude. You will use a Pasco Rotary Motion Sensor, the U of T Data Acquisition Device (DAQ), and the RMS program. The RMS program is based on the SignalExpress software platform from National Instruments.
The Rotary Motion Sensor is mounted on a support, and has a rod with a mass on it mounted on it. The rod and mass will be the physical pendulum you will study. Also part of the rotating system are the plastic disc on which the rod is mounted and the axel connected to the Rotary Motion Sensor.
Setup
There are two phone plugs on cables connected to the Rotary Motion Sensor. These are plugged into the corresponding terminals on the Data Acquisition Device. On the left side of the Device are two pairs of terminals labeled Digital Channels; each pair has one labeled with a yellow circle and the other with a black circle. The corresponding yellow and black plugs from the sensor are plugged into pair labeled 0, yellow to yellow and black to black.
When you start RMS the position of the Rotary Motion Sensor defines the angle to be zero. Make sure that the pendulum is stationary and start RMS.
Occasionally the hardware and software gets confused about the resolution of the measurement.
1. Rotate the pendulum by one full rotation. The angle on the graph should read 6.28 radians (= 360º) either a positive or negative number.
2. If the angle is not correct, disconnect the Rotary Motion Sensor from the DAQ the plug it back in.
3. Stop and restart the RMS program and check that one full rotation reads 6.28 radians.
4. Stop RMS.
Trial Run
It is a good idea to do a trial run first. You should do a run with a maximum amplitude [pic].
Make sure the pendulum is stationary.
Start RMS.
Rotate the pendulum to some about 1 radian. You can read the angle on the graph.
Release the pendulum. The graph will show the angle of the pendulum as a function of time.
After about 10 oscillations stop RMS.
Right click on the graph. Choose Visible Items / Cursors.
Drag the solid cursor to one of the early oscillations. The horizontal line of the cursor will track the data.
Drag the dotted cursor to one of the later oscillations.
At the bottom of the graph the Dx field is the time between the two cursors. y1 and y2 are the angles of the first and second cursors respectively, in radians, and Dy is the change in the angle.
Thus you may calculate the value of the period and the maximum amplitude.
Note: due to a mis-feature of SignalExpress, sometimes when you take another data set it is difficult to use the cursors. To fix, turn the cursors off and then back on using Visible Items / Cursors.
There are some issues that should be considered:
A. The time resolution of the software is 0.05 s. You can confirm this by moving one of the cursors the minimum possible amount and seeing how much the value of the time changes. Considering that the measured value of the time when the pendulum is at maximum amplitude is unlikely to be the exact value when it actually occurred, a reasonable value for the error in the measurement of the time Δt is perhaps ± 0.03 s. Is this a reasonable estimate? If yes, explain why in your own words. If no, what is a more reasonable value?
B. If you position the cursors at successive maximum values of the amplitude, then the time between them is the period T of oscillation. What is a reasonable estimate of the uncertainty in the period, ΔT?
C. Say you position the cursors for n oscillations. Call the final value of the time tf and the initial value ti. The software shows the value of tf - ti as Dx at the bottom of the window. The value of period is T = Dx/n. Now what is the uncertainty in the period, ΔT? Is this a better way to determine the period?
D. Since this is not a perfect frictionless system, the maximum amplitude decreases with time. You can see this in your data. Thus the value of θmax is not well defined. What is a reasonable estimate of the error in its value? Why? How does the value depend on the number of oscillations between the cursors?
E. Considering the results of Part C and Part D, how many oscillations should you use to determine the values of the period for a given maximum amplitude? Note that there is a trade-off: increasing the number of oscillations n reduces the error in the period but increases the error in the maximum amplitude. Explain your choice.
Data Collection and Analysis
Collect data for the period for a number of values of the maximum amplitude.
Use CreateDataSet to create a dataset of values for the period and maximum amplitude, including their errors, and save it into your Team’s area. You will want the values of the maximum amplitude to be the Independent (x) Variable, and the values of the period be the Dependent (y) Variable.
You can then use ViewDataSet to view your data. You may wish to print the window and staple it into your lab book.
Does the data look reasonable? Does it appear to be consistent with the values of Table 1?
Parameterising the Data
You have determined the period of the pendulum for measured values of the maximum amplitude. Here we explore how to use that data to determine the value of the period for maximum amplitudes that have not been measured.
The relation between the period and the maximum amplitude is a complicated function f() involving elliptical integrals:
[pic] (12)
However, it might be reasonable to approximate the function as a polynomial:
[pic] (13)
Can you eliminate the odd coefficients a(1), a(3), etc. from the series using a physical argument? If yes, what argument can you use?
Use the PolynomialFit program to find the best fit the data. Do not include any terms that are physically unreasonable. You will find that adding some terms to the polynomial will not improve the quality of the fit: your goal is to find the minimum number of terms in the polynomial that provides a good fit to the data. Any fitted parameters a(i) that are zero within errors should not be included in the fit.
From the best fit, what is the value of T0?
How well does the best fit do in duplicating the values of Table 1? Does it do a better job for small angles than for large ones?
Activities 9 and 10 were concerned with pendulums whose motion was approximately but not exactly Simple Harmonic. In Activities 11 and 12 we will investigate some periodic systems whose motion is not even close to being Simple Harmonic. Both Activities are only brief looks at the two systems, and are just for your interest.
Both systems that are investigated are chaotic. Here are some characteristics that all chaotic systems have:
No analytic formula can even approximate the motion.
The motion will never repeat. Ever!
If two identical periodic systems are started with almost identical positions and speeds, soon their motions will be radically different from each other.
Chaotic systems are deterministic. If they start with exactly identical initial conditions their subsequent behavior will be exactly the same.
Characteristic 3 is sometimes called the butterfly effect. This is because if history were a chaotic system then the outcome of World War II could have been determined by whether or not a butterfly landed on a particular flower in the Himalaya Mountains in 1848. You may learn more about chaotic systems from:
Activity 11
Our first example of a non-harmonic pendulum is the double pendulum. It is an example of a chaotic system.
Start the IDLE for VPython program. Use File / Open … to open the doublependulum.py program in the examples directory; this directory is the default one that is opened when IDLE is first started. Run the program. You are welcome to look at the code for this program, but will wish to know that it is written in terms of a sophisticated form of Newton’s Laws called a Lagrangian formulation.
You may investigate the double pendulum further with a Java applet by Peter Selinger at Dalhousie University, Halifax Nova Scotia:
From the trace of the trajectory you can see characteristic #2 above illustrated. By clicking on the Restart button you can see characteristic #4 demonstrated.
Activity 12
In Activity 4 you may have used a Motion Sensor to track the position of a mass oscillating up and down on a spring. If the support of the spring is oscillating up and down, for some frequencies and amplitudes of oscillation this system too is chaotic.
FEED ME.
This Student Guide was written by David M. Harrison, Dept. of Physics, Univ. of Toronto in May 2008
Last revision: November 8, 2008.
Activity 3.A is based on Activity 14.1 Part 3 of Randall D. Knight, Student Workbook with Modern Physics (Pearson Addison-Wesley, 2008). Activity 6 is based on Activities 1.43 Parts 10, 11, and 12 of Knight’s Student Workbook. Activities 7 are 8 are from David Harrison and William Ellis, Student Activity Workbook, 3rd ed. (Norton, 2008), Activities 15.8, 15.9.
Appendix 1 – Listing of Pendulum.py
# Solve the pendulum using numerical approximation
# Copyright (c) 2008 David M. Harrison
# The next line is an internal revision control id:
# $Date: 2008/05/10 10:33:45 $, $Revision: 1.1 $
# Import the visual library
from visual import *
# The initial angle in radians.
theta = pi/2.0
# The initial angular velocity
omega = 0
# Set g and the length of the pendulum
g = 9.80
L = 1.00
# These four lines control the size of the window of
# the animation and the scale. The details of these lines
# are not important for our purposes.
scene.autoscale = 0
scene.height = 600
scene.width = 600
scene.range = vector(2.0,2.0,2.0)
#
# Now we build the pendulum which we will animate.
#
# The support for the pendulum
support = cylinder( pos = (0, 0, -0.5), axis = (0,0,1), radius = 0.02)
# The "frame" construct groups two or more objects into a single one.
# Here we group the cylinder and the sphere into a single object
# which is the pendulum.
pendulum = frame()
cylinder(frame=pendulum, pos=(0,0,0), radius=0.01, length=1, color=color.cyan)
sphere(frame=pendulum, pos=(1,0,0), radius=0.1, color=color.red)
# Position the pendulum.
pendulum.pos = (0,0,0)
# Rotate the pendulum about the z axis. Note that VPython measures
# angles with respect to the x (horizontal) axis. We are measuring
# angles with respect to the vertical (-y axis) so we subtract
# pi/2.0 radians from the angle.
pendulum.rotate(axis = (0,0,1), angle = theta - pi/2.0)
# The time
t = 0.
# Below we will want to store the old value of the time.
# Set it the "impossible" value of -1 initially.
t_old = -1.
# The time step
dt = 0.0005
# The value "1" is equivalent to true. So this causes the while
# loop to run forever.
while 1:
# Set the rate of the animation in frames per second
rate(1/dt)
# The angular acceleration, i.e. the second derivative of the
# angle with respect to time.
alpha = -(g/L) * sin(theta)
# The new value of the angular velocity
omega = omega + alpha * dt
# The change in the angle of the pendulum
d_theta = omega * dt
# A rough and ready way to estimate the period of the oscillation.
# It the angle is positive and adding d_theta to it will make
# it negative, then it is going through the vertical
# from right to left.
if(theta > 0 and theta + d_theta < 0) :
# If t_old is > 0, then this is not the first cycle of
# the oscillation. The difference between t and t_old
# is the period within the resolution of the time step dt
# and rounding errors. Print the period.
if(t_old > 0):
print "Estimated Period =", t - t_old, "s"
# Store the current value of the time in t_old
t_old = t
# Rotate the pendulum about the z axis by the change in the angle
pendulum.rotate(axis = (0,0,1), angle = d_theta)
# Update the value of the angle
theta = theta + d_theta
# Update the time
t = t + dt
Numerical Approximation Module
Student Guide
Concepts of this Module
• Introduction to the Python programming language.
• Numerical approximation as an alternative to analytic solutions.
Introducing Python
Here we briefly introduce the Python language and some of the programming constructs that will be used in the main part of this Module.
The Python programming language is free and open source, with a huge community of developers. Although it is an ideal first language to learn, you may wish to know that it is not a “toy”. It is used extensively by Google, NASA, the Large Hadron Collider just being lit up in Switzerland, Youtube, Air Canada, and many more.
Traditionally the first computer program simply prints hello, world. Here is a complete Python program that does this:
print "hello, world"
Here is another complete program that also prints hello, world:
what = "world"
print "hello,", what
The first line of this program assigns world to a variable named what. The next line then prints hello, followed by whatever the variable named what is set to, world in this case. The Python interpreter executes the lines of this “program” in order.
Today we will wish to have Python execute some lines of the program over and over again. We will use a while loop to do this. This loop has the form:
while something_is_true:
execute this line of the program
then execute this line of the program
then execute this next line of the program
After executing the third line after the while statement, it goes back to the while statement: if something is still true then it executes the following lines again, and so on.
We have prepared a program named LoopDemo.py which demonstrates this loop. Here is a listing of the program.
Listing of LoopDemo.py
# All lines like this one that begin with a "#"
# are comments. All other non-blank lines are
# program statements.
# Set a variable named "x" to a value of 0
x = 0
while x < 3:
print x
# Increase the value of x by one.
x = x + 1
# End of the while loop. Go back to
# the while statement again.
You may wish to know that the lines following the while statement must be indented as shown.
Start the IDLE for VPython program. Use File / Open … to open the file LoopDemo.py which is located in Feynman:Public/Modules/NumerApprox folder.
Predict what will happen when this program is run.
Check your prediction by running the program: use Run / Run Module or press the F5 key on your keyboard.
Sometimes we wish to use a while statement to have the program execute the same lines over and over until it is manually stopped. The LoopDemo2.py file in the same directory does exactly this. A listing of this program is in Appendix 1.
Predict what will happen when this program is run.
Check your prediction by running it.
Also in the Feynman:Public/Modules/NumerApprox folder is the file LoopDemo3.py, and a code listing is in Appendix 2. It differs from LoopDemo2.py in two ways:
1. The first print t statement is removed.
2. Inside the while loop the two statements that increment the value of the time and prints the value of the time are reversed.
Predict what will happen when this version is run. Check your prediction by opening the file and running it.
The Spring-Mass System and Numerical Approximation
For a mass m on a spring with spring constant k Newton’s Second Law is:
[pic] (1)
This is a second-order differential equation, and if one knows enough calculus one can solve it to get:
[pic] (2)
where:
[pic]
But if one doesn’t know enough calculus or just doesn’t want to bother with a differential equation, a moderately powerful computer provides a nice alternative. The basic idea is that we will start with the mass at some known position and calculate its acceleration, how fast it is moving and where it will be small timestep Δt later, and keep doing this over and over again. Here is how one may do this numerical approximation:
1. From the mass’ current position x we can calculate the acceleration a of the mass: [pic]
2. If the speed of the mass is v, then calculate a new speed vnew = v + a Δt.
3. If the position of the mass is x, calculate a new position xnew = x + vnew Δt.
4. Go back to Step 1 and repeat.
Of course, this method is just an approximation. However given a sufficiently powerful computer to do the calculations we can make the approximation as close to correct as we wish by making the timestep Δt sufficiently small.
We have prepared a Visual Python (VPython[3]) animation which both uses Eqn. 2 and implements the numerical approximation described above.
The Activity
A. Open the IDLE for VPython program. Use File / Open … to open the file SHM.py which is located in Feynman:Public/Modules/NumerApprox. Use Run / Run Module or press the F5 key on your keyboard to start the animation. The upper yellow sphere uses Eqn. 2, and the lower green sphere uses the numerical approximation. Can you see any differences between the motions of the two spheres? For fun you may wish to know that:
• Holding down the right mouse button and moving the mouse allows you to rotate the view of the animation.
• Holding down both mouse buttons and moving the mouse up or down allows you to zoom in and out on the animation.
B. For your convenience a listing of the SHM.py code is included in Appendix 3 of this document. In the Feynman:Public/Modules/NumerApprox folder the file CodeBig.pdf also lists the code using big fonts; you may wish to print this file and place the pages on the whiteboard using small magnets. Including empty lines there are 90 lines in the file. How many of them are program statements?
C. Some lines of the code are used only for the animation of the yellow ball; some lines are only for the animation of the green ball; some lines are shared for the animations of both balls; still other lines are commands to control the animation speed, set up the calculation loop, or set the “stage” for the animation. Circle or use a highlighter on all the lines in the code that are used only for the animation of the yellow sphere and label them with Y; if a yellow highlighter is available it would be a good choice for this.
D. Preferably using a different color pen or highlighter, circle or highlight all the lines in the code that are used for the animation of both spheres and label them with B.
E. Describe in your own words how the program animates the motion of the yellow ball.
F. From the parameter values set in the code calculate the period T of the oscillation. Does your calculated value match the actual period you see in the animations?
G. About 60% down the code listing the maximum amplitude of the motion ampl is calculated. Did you circle this in Part C? If not, should you have? Is the calculation correct? (Hint: think about conservation of energy.)
H. Preferably using a third color pen or highlighter circle or highlight all the lines in the code that are used only for the animation of the green sphere and label them with G; a green highlighter would be ideal if available. Circle or highlight all the lines that control the animation speed, set up the calculation loop, or set the “stage” for the animation, and label them with C; a fourth color pen or highlighter would be nice if possible. Follow the code for all the lines that are used for the animation of the green sphere. Does it surprise you that nowhere in these lines of code does a trig function appear? Explain.
I. In the code for the yellow ball, the value of the time is incremented and then the new position of the ball is calculated. Is this correct? What if those two lines were reversed?
At the end of this Module, you will want to staple your “de-constructed” code into your lab book.
For the Keen
Here are some things you may wish to do. They are not intended to be part of the Activity of this Practical, but instead some things you may wish to explore on your own.
Some systems, particularly chaotic ones, are not analytically solvable: there is no equation that describes the motion. For such systems numerical approximation is the only way that they may be studied. When VPython first starts, using the File / Open … command lists the examples that are shipped with the software. The doublependulum.py program in that directory is an example of a chaotic system which is not analytically solvable but here is solved by numerical approximation. The physics behind this animation is fairly formidable, but the basic idea is the same as the SHM.py code you used here. There are many other interesting examples that are shipped with the software.
You may also save a copy of the SHM.py file and try modifying it by changing some of the parameters set in the code. You will want to know that by default every time you run the program VPython first saves the code into the file. Thus you may wish to consider working on a copy of the master file, named perhaps SHM_work.py.
One simple change you could make to SHM.py involves efficiency. As written determining the yellow sphere’s position involves calculating the angular velocity sqrt(k/mass) for every iteration of the loop. Calculating the value once before entering the loop and then using the calculated value would mean that the program has to perform many less calculations.
Appendix 1 – LoopDemo2.py Code Listing
# All lines like this one that begin with a "#" are
# comments. All other non-blank lines are program
# statements.
# Import the visual library.
from visual import *
# Set the time
t = 0
# Set the timestep
dt = 1
# Print the current value of the time
print t
# The next line causes the indented lines that follow
# it to be repeatedly executed in the loop. The construct:
# 1==1
# means "is one is equal to one?" which is always true.
# Thus double equal signs like this mean something different
# than a single equal sign, such as is used above to set the
# values of the time and the timestep.
while 1==1:
# Do one calculation every second
rate(1)
# Increment the value of the time and print the result.
# Here the single equal sign means set the value of t to
# whatever appears to the right of the equal sign.
t = t + dt
print t
# End of the while loop. Go back to the rate(1) statement
# and start over.
Appendix 2 – LoopDemo3.py Code Listing
# All lines like this one that begin with a "#" are comments.
# All other non-blank lines are program statements.
# Import the visual library.
from visual import *
# Set the time
t = 0
# Set the timestep
dt = 1
# The next line causes the indented lines that follow
# it to be repeatedly executed in the loop. The construct:
# 1==1
# means "is one is equal to one?" which is always true.
# Thus double equal signs like this mean something different
# than a single equal sign, such as is used above to set the
# values of the time and the timestep.
while 1==1:
# Do one calculation every second
rate(1)
# Print the time and then increment its value.
# Here the single equal sign means set the value of t to
# whatever appears to the right of the equal sign.
print t
t = t + dt
# End of the while loop. Go back to the rate(1) statement and
# start over.
Appendix 3 – SHM.py Code Listing
# All lines like this one that begin with "#" are comments.
# All other lines are program statements.
# The next line is an internal revision control id:
# $Date: 2007/11/08 17:19:19 $ $Revision: 1.2 $
# Copyright (c) 2007 David M. Harrison
# Import the visual library.
from visual import *
# These four lines control the size of the window of
# the animation and the scale. The details of these lines
# are not important for our purposes.
scene.autoscale = 0
scene.height = 400
scene.width = 800
scene.range = vector(60, 60, 60)
# Create the green ball that will execute simple harmonic motion
# by numerical integration.
greenBall = sphere (color = color.green, radius = 2)
# yellowBall will execute simple harmonic motion using a sine function.
yellowBall = sphere (color = color.yellow, radius = 2)
# The initial x position of the balls: this is
# the equilibrium position.
x = 0
# Position the balls. pos is a built-in of VPython, and
# lists the (x,y,z) coordinates. The x axis is horizontal,
# y axis is vertical, and the z axis is perpendicular to
# the plane of the screen. We place the green ball just
# below the center of the scene, at y - -10.
#
greenBall.pos = (x,-10,0)
# yellowBall is above the first ball: it's y coordinate is 10,
# just above the center of the scene.
yellowBall.pos = (x, 10, 0)
# The initial x component of the velocity of the balls:
# all other components are zero.
vx = 150
# The spring constant
k = 9.0
# The mass of the balls
mass = 1.0
# The amplitude of yellowBall's motion
ampl = sqrt(mass/k) * vx
# The time
t = 0
# This is the time step
dt = 0.005
# This causes the following indented lines
# to be executed forever in a loop.
while 1 == 1:
# Set the rate of the animation
rate(1/dt)
# The acceleration in the x direction.
a = -(k/mass) * x
# Update the speed using the acceleration. Note
# that we "recycle" the variable vx, replacing the
# old value with the new one.
vx = vx + a*dt
# Update the x position of the ball using the speed.
x = x + vx*dt
# Position greenBall at the new x position
greenBall.pos = (x, -10, 0)
# Update the time
t = t + dt
# Now we calculate simple harmonic motion using
# a sine function and position yellowBall using the result
# of the calculation
x2 = ampl * sin( sqrt(k/mass)* t)
yellowBall.pos = (x2,10,0)
This Guide was written by David M. Harrison, Dept. of Physics, Univ. of Toronto in November 2007.
Last revision: March 4, 2008.
Fluids Module
Student Guide
Concepts of this Module
• Fluids
• Pressure
• Buoyancy
• Fluid dynamics
The Activities
Activity 1
Open the gas-properties.jar animation which is located at feynman:public/Modules/Fluids. There are many useful ways to use this animation, and we will only draw you attention to a couple of things that you may wish to do; you are encouraged to explore further.
Here is a screen shot of the default animation after some Heavy Species molecules have been pumped into the container:
[pic]
A. You will notice that the reading of the Pressure gauge is not constant. Explain why this is so. What would be necessary for the pressure reading to be more constant? How would you present a value for the pressure that also expresses your observed variations?
There are many options for controlling the animation. We shall describe two of them.
[pic]
1. By default the acceleration due to gravity g is zero. You may introduce a non-zero value of g with the Gravity slider.
2. By clicking on the Measurement Tools button you may turn on the Layer tool. This tool measures the pressure in the gas at a specified height; you may drag the position of the measurement with the mouse. You can also specify the time over which the value of the pressure is averaged.
Here are some suggested explorations.
B. Use the Layer tool with various settings of the Averaging Time. Describe what happens. If this was not part of your answer to Part A, should it have been?
C. With Gravity set to 0, predict how the pressure in the gas varies with height. Check your prediction using the Layer tool. Were you correct? If the pressure varies with height, does it vary is the height, the height squared, one over the height, or what?
D. Introduce a non-zero Gravity. Predict how the pressure varies with the height. Check your prediction. Were you correct? If the pressure varies with height, does it vary is the height, the height squared, one over the height, or what?
Activity 2
Cylinder A is being filled to the level shown. As the water is added to the cylinder it flows along the horizontal pipe and up B, C and D, which are all open at their tops.
[pic]
Rank the heights of the water in A, B, C, and D when A is filled. Check your prediction using the supplied apparatus. Was your prediction correct? If yes, what physical principles did you use to make a correct prediction? If no, explain the actual result.
Activity 3
A rigid rectangular container filled with water is at rest on a table as shown. Two imaginary boundaries divide the water into three layers of equal volume. No material barrier separates the layers.
A. Draw a free body diagram for each layer. The label for each force should include:
• A description of the force, and
• The object on which the force is exerted, and
• The object exerting the force.
B. Rank the magnitude all the vertical forces you have drawn for Part A, from the smallest to the largest. Explain how you determined the ranking.
C. Rank the magnitude of all the horizontal forces you have drawn for Part A, from the smallest to the largest. Explain how you determined the ranking.
Activity 4
A small square hole of area A is cut in the side of the container of Activity 2. The centre of the hole is a height z above the tabletop. Consider the rectangular section of water of area A aligned with the hole, as shown.
A. Draw a free body diagram of all the forces acting on the rectangular section of water.
B. What will happen to the water just inside the hole?
Activity 5
A bucket of water has a spring soldered to the bottom. Attached to the other end of the spring is a cylindrical cork of mass m, height h and area A which is stationary below the surface of the water, as shown. The top of the cork is a depth d below the surface of the water. The spring has a spring constant k and is stretched a distance x from its equilibrium position. The density of the water is ρ.
Draw a free body diagram of all the vertical forces acting on the cork. Evaluate the magnitude of those forces. Determine x, the amount that the string is stretched from its equilibrium position.
Activity 6
A hydrometer measures the density of a liquid. They are widely used to measure the alcohol content in the brewing of beer, the electrolyte content of battery acid, and more.
The device is placed in the liquid whose density is to be measured, and the density is read by the place on the scale where the surface of the liquid touches the stem.
Here is a close-up figure of two possible ways that the markings on the scale of the hydrometer can be arranged. Which of these arrangements are correct? Explain.
[pic]
Activity 7
Please do this Activity with all the apparatus in the supplied dishpan to minimize the water spilled onto the tabletop.
You are supplied with a beaker. You should fill it with water nearly to the top. Place the supplied medicine dropper in the water with the squeeze bulb on top. Suck enough water up into the medicine dropper that it just barely floats.
You are supplied with an empty 2 liter plastic pop bottle. Fill it to the brim with water. Transfer the filled medicine dropper to the water in the pop bottle.
Screw the top tightly on the bottle. Squeeze the bottle. What happens to the medicine dropper? What happens when you quit squeezing the bottle? Explain why squeezing the bottle and increasing the pressure of all the fluids within would cause the observed motion. This is called a Cartesian diver.
The supplied toothpicks make it easy to “fish” the medicine dropper out of the bottle.
When you are finished with this Activity, carefully empty all the water into the sink.
Activity 8
You may have noticed that the bubbles in a glass of a carbonated beverage (soda, beer, champagne, etc) accelerate as they rise from the bottom. Explain.
Activity 9
A ship is in a canal lock, which is only a little bit larger than the ship itself. The ship is loaded with steel ingots, which are large bars of steel. The crew becomes angry with the captain of the ship and throws the steel ingots overboard into the water of the lock.
Does the level of the water in the lock rise, lower, or stay the same?
Check your prediction. Place the supplied plastic tank in the dishpan and fill the tank about half-way with water. Place the supplied weight in the bottom of the supplied plastic boat and gently place it in the water. You may mark the height of the water in the tank with a small piece of masking tape. Carefully lift the boat out of the water, place the weight at the bottom of the tank, and put the boat back in the water.
When you are finished with this Activity, carefully empty the water into the sink.
Activity 10
A water tank with water of height h has a small hole cut in the side at height z. The water strikes the ground at x. The figure shows the streamline from the top of the water at A to just outside the hole B. Recall that Bernoulli’s equation is:
[pic]
If the hole is small, it is reasonable to approximate that the speed of the water at A is zero. Since point A and B are in contact with the outside air, it is reasonable to approximate that the pressure is the same at point A and B, that of atmospheric pressure in the room.
A. What will be the shape of the stream of water emerging from the hole until it strikes the ground?
B. Without using any equations, describe how the speed of the water at B varies with z, How will the distance x depend on z?
C. Use Bernoulli’s equation and your knowledge of projectile motion to derive the answers to Part B. For what value of z will x be a maximum? What approximations are you making? Are those approximations reasonable?
D. You are supplied a tank with small holes cut in it at values of z = 0.75 h, 0.50 h, and 0.25 h, where the height of the water h is indicated by a mark on the tank. Place the tank in the supplied dishpan: place it on one end of the dishpan with the holes pointing towards the other end of the dishpan. Fill the tank with water to the mark. As the water level drops appreciably add water. Is what you see consistent with your results from Parts B and C?
When you are finished with this Activity, carefully empty the water into the sink.
Activity 11
When an object falls through a fluid, either a liquid or a gas, there are three forces that act on it:
1. The downward force due to gravity, [pic]. This is the weight of the object.
2. An upward buoyant force, [pic]. As Archimedes realized over 2,000 years ago, this is equal to the weight of the fluid that the object displaces.
3. An upward drag force, [pic].
In this Activity we will concentrate on the drag force exerted on a sphere falling through a fluid. We assume that the surface of the sphere is perfectly smooth. We will use the following variables in the discussion:
• r: the radius of the ball.
• v: the instantaneous speed of the ball.
• ρ: the density of the fluid.
• η: the kinematic viscosity of the fluid. This is sometimes called liquid friction. It is measured in units of pressure × time.
Here are some values for the density and viscosity of various fluids.
|Fluid |Density (kg/m3) |Viscosity (mPa-s) |
|Superfluid |-- |0 |
|Air (20 ºC) |1.2 |0.0182 |
|Water (20 ºC) |998 |1.00 |
|Olive Oil (88 ºC) |914 |43.2 |
|Glycerine (20 ºC) |1260 |658 |
|Honey (20 ºC) |1,500 |5,000 |
There are various ways that the fluid can flow around the sphere. If the speed of the ball is small, the flow is “smooth” or “laminar”. In this case it turns out that the drag force is proportional to the speed.
[pic]
This was first shown by Stokes in 1851. This will be explored further in Part E.
When the ball is going faster, turbulence develops in the fluid behind the ball. In this case the drag force is approximately proportional to the speed.
[pic]
Note that in both of these cases, as the ball’s speed increases the drag force increases. Thus at some point the drag plus buoyant forces approaches the magnitude to the weight of the ball, so asymptotically there is zero net force acting. Then the speed of the ball becomes constant: the value of the speed is called the terminal velocity. For a sky diver falling face down with arms and legs outstretched, the terminal velocity is about 55 m/s. If the sky diver falls feet first, feet together and arms close to the body, the terminal velocity goes up to about 90 m/s. When the sky diver opens the parachute, the drag force goes way up and the terminal velocity falls to about 5 m/s.
Whether the fluid flow around the ball is laminar or turbulent turns out to depend only on a single dimensionless number, the Reynolds number Re.
[pic]
|Re |Type of Flow |
|≤ 1 |Laminar |
|1 - 1 × 103 |Transition |
|1 × 103 – 1.5 × 105 |Turbulent |
Note that for constant r, ρ, and η the Reynolds number is proportional to speed. Therefore when a ball is dropped from a large height, Re increases until the terminal velocity is reached.
When the Reynolds number reaches ~ 1.5 × 105 the forces on the fluid near the ball become extreme, and both the wake and the layer of fluid right next to the ball become turbulent. This causes a sudden change in the way the luid flows around the ball, and the turbulent wake becomes narrower. When this happens, the drag force drops and the acceleration of the ball increases. This is called the drag crisis. As the speed increases further, the drag force resumes increasing with speed.
A. A ball of radius r is falling through a fluid and at some time has an instantaneous speed v. A second ball of radius 2r is falling through the same fluid. At what instantaneous speed will the second ball have the same flow pattern of fluid around it as the first ball?
B. Here is the URL of a Flash animation of dropping a ball from the CN Tower:
The above link is to a fixed size animation which works nicely if only one person it viewing it. For use in the Practical itself a version which can be resized to be larger so that the entire Team can see it is better. Here is a link to such a version:
Open the animation and explore how it works.
For the keen some details about this animation and the one you will explore in Part D are in the Appendix.
C. For air, in SI units the Reynolds number is:
[pic]
For the billiard ball, 5-pin bowling ball, and 10-pin bowling ball calculate the speed for which the drag crisis occurs. Are these results consistent with what you see in the animation of Part B?
D. Here is the URL of a Flash animation of dropping a ball in a liquid:
As with the animation of Part B, you may access a resizable version at:
Open the animation and explore how it works.
E. For small Reynolds numbers, so the fluid flow is laminar, the drag force is:
[pic]
You may be surprised by the fact that the density of the fluid does not appear in this equation. When the sphere is at terminal velocity the net force is zero:
[pic]
Therefore:
[pic]
For the animation of Part D, set the following values:
• r = 20 mm
• η = 5850 mPa-s
• ρliquid = 1500 kg/m3
• ρball = 5000 kg/m3
Note the values of the ball weight and buoyant force and their difference. Run the animation and note the terminal velocity.
Now set the radius of the ball to 25 mm. Adjust the viscosity of the liquid so the drag force will be the same as the previous case. Adjust the liquid and ball densities so the ball weight minus the buoyant force is about the same as the previous case; you are unlikely to find values of the densities which are exactly the same, but can find ones that make the value almost the same. Does the animation match the theory? In particular is the motion of the ball the same for these two cases?
Appendix
Although the details of how the animations of Activity 10 work are not important for your learning of fluid dynamics, here we “lift the hood” to discuss the internals of the animations.
Except in the limit of laminar flow, the theory of drag forces is not easily solvable. Thus the animation uses a mixture of experimental data and some heuristic formulae that describe the data reasonably well. It turns out to be convenient to describe the drag force in terms of a drag coefficient CD(Re), which is a function only of the Reynolds number. Then the drag force is:
[pic]
For laminar flow (Re ≤ 1):
[pic]
For larger values of the Reynolds number, experimental data on the dependence of the drag coefficient on the Reynolds number must be used. The data used in the animation is adapted from H. Edward Donley, UMAP Journal 12(1), 47 (1991), .
To parameterize this data involves some truly ugly equations. We used forms by John Versey and Nigel Goldenfleld, “Simple viscous flows: From boundary layers to the renormalization group”, Rev. Mod. Phys. 39(3), 883 (2007), .
The Donley data and the Versey and Goldenfleld interpolation of it is shown on the next page.
The turbulent flow case (1 × 103 < Re < 1.5 × 105 ) corresponds to the part of the above plot where the drag coefficient is approximately constant independent of the Reynolds number. Note that, despite the notation used in the axes of the figure, the values are the natural logarithms of the values.
The drag crisis is when the drag coefficient suddenly drops.
Once the drag force for a given speed has been determined, then we know the net force acting on the ball and hence its acceleration. We use a numerical approximation to find
the motion of the ball. The method is similar to one you may have explored in the Numerical Approximation Module. For a time step dt.
[pic]
1. From the acceleration, calculate the new speed of the ball:
vnew = vold + a × dt
2. From the new speed calculate the new position of the ball:
ynew = yold + vnew × dt
3. From the new value of the speed calculate the new drag force and then the new acceleration of the ball.
4. Go to 1 and repeat.
The above scheme turns out to not be accurate enough for our animation, so an extension of it called a 4th order Runge-Kutta is used. It turns out that for this calculation to be stable we must iterate the Runge-Kutta 10 times for every frame of the animation. Since the animation runs at 12 frames per second, this means that the time step dt is 1/120 = 0.17 s.
This Student Guide was written by David M. Harrison, Dept. of Physics, Univ. of Toronto, May 2008.
The animation used in Activity 1 is from the Physics Education Technology (PhET) group at the University of Colorado, . Activity 2 is based on Lillian McDermott et al., Tutorials in Introductory Physics (Prentice Hall, 20020, ST 219. Activities 5 and 7 are based on David M. Harrison and William Ellis, Student Activity Workbook, 3rd ed. (Norton, 2008), 18.4 and 18.6. The figure for Activity 5 is slightly modified from a figure from Wikipedia, , retrieved June 19, 2008.
Last revision: November 20, 2008.
-----------------------
[1] 1 radian = 57.2958°, or 2π radians = 360°, or π radians = 180°.
[2] These values were calculated with Mathematica.
[3] VPython is free, o数潳牵散湡癡楡慬汢潦楗摮穯ⱥ䴠捡楌畮⁸湡乕塉映潲–奈䕐䱒义⁋栢瑴㩰⼯睷灶瑹潨牯⽧•ᐁ瑨灴⼺眯睷瘮祰桴湯漮杲ᔯമ̍ഄ̍ഄግ䅐䕇†ക倓䝁⁅ᐠ㠶ക䡐ㅙㄳ倠慲瑣捩污慍畮污††††††湉牴摯捵楴湯䡐ㅙㄳ倠慲瑣捩污慍畮污pen source, and available for Windoze, Mac, Linux and UNIX from .
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.