GEOG 1F91 Lab Assignment #5 (Winter 2024) Student Handout Final
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GEOG 1F91 Principles of Physical Geography Lab Assignment #5 –
Angle of Repose and Angle of Internal Friction - Winter 2024 TEXT REFERENCE:
Chapter 14 –
Rock Structure, Weathering and Mass Wasting Chapter 17 - The Work of Wind and Waves OBJECTIVE: This lab illustrates how sediment behaves upon forming a slope, and how this relates to slopes observed in the natural environment. Why is this important? In considering the natural environment, we must ask ourselves why some slopes are able to maintain near vertical angles (e.g., the Niagara Escarpment, including Niagara Falls, and the Grand Canyon) while others are not and fail, sometimes dramatically (see Fig. 6.1, at right). Materials comprising the Earth’s surface tend to have specific characteristic properties that govern their behaviour, including the maximum slope angle they are able to maintain. This is not to say that any given material will always display this angle in the field as various processes are at work to, usually, reduce this angle (although in some cases, maximum slope angles can be temporarily increased). While these adjustments in slope angle are usually slow, gradual processes, they can sometimes be catastrophically rapid. This lab endeavours to illustrate how materials behave as they begin to fail downslope, and how the maximum angle is attained. INTRODUCTION The Earth’s surface is composed of slopes, and the angle of these slopes is, for the most part, below that which allows particles to move down them. The forces at work on a slope are illustrated in Figure 6.2. All movement is the result of an energy gradient and in this case it’s the difference in potential energy (the energy of an object by virtue of height) between the top and bottom of the slope. While we think of the force of gravity (W) causing objects to roll downhill, one must consider that gravity acts directly downward rather than downslope. As such, we must resolve the downward gravity force into two components: the shear force (T) acting down the slope and the normal force (N) acting normal (meaning at right angles) to the slope. The strength exhibited by the material against the force of gravity (or its Figure 6.1: Catastrophic Mass Movement Source: Pidwirny, M. (2006). "Hillslope Processes and Mass Movement". Fundamentals of Physical Geography, 2
nd
Edition
. http://tinyurl.com/ohahfn4
two components, T and N) is made up of several components, including friction between individual particles as they try to slide past one another and the degree of interlocking of the individual particles comprising the material. Together, these are often referred to as the internal friction of the material. So long as the strength resisting movement exceeds the downslope forces, the slope will remain stable. Figure 6.2: The Forces on a Slope It is obvious that as a slope becomes steeper, movement is more likely, but why is this so? The conditions in Figure 6.2 can be redrawn using different slope angles to illustrate this. At relatively shallow angles (Figure 6.3a) it can be seen that, when the gravity vector (W) is resolved into its two components, the normal force (N) is relatively large as compared to the shear force (T). As the slope angle increases, resolving the same gravity vector results in an increasingly greater shear force compared to the diminishing normal force (Figure 6.3b). Figure 6.3: The Forces of Slopes of Different Steepness
Resisting this is, of course, the internal friction of the material itself. When shear force becomes large enough to overcome the internal friction, material begins to move downslope. Movement will also occur when environmental conditions operate to reduce the internal friction (the friction between the particles themselves or between the particles and the surface on which they rest). For example, high soil moisture levels can cause slope failure by effectively reducing the frictional forces between soil particles, while lower soil moisture levels can actually enhance stability via negative pore water pressures. The addition of sediment onto an existing slope can also lead to failure, due to the increase of gravitational force over the frictional force. Downslope movement is typically characterized by two different, but related, measures. The maximum angle that unconsolidated, non-cohesive material can maintain when poured into a pile is known as the angle of repose (see Fig. 6.4, at right). Conversely, the angle of internal friction is measured by slowly increasing the slope angle and noting when material begins to slide downslope. In both cases the frictional forces maintaining the slope are being counteracted by gravitational forces (resolved into its normal and shear components) trying to reduce the slope angle. As such, one would expect these two angles to be the same but this is often not the case. This apparent discrepancy is the phenomenon we will investigate in this lab where we will examine experimentally both of these two situations: (1) The angle of repose: where sediment is poured into a conical heap and we examine the surface angle attained by the material, and (2) The angle of internal friction: the angle at which particles will begin to slide down slopes of comparable roughness. The measured results from these experiments must be carefully recorded and will be used as the basis for your conclusions. You should also observe all factors surrounding the experiments as a basis for assessing the validity of the results, and for suggesting improvements. Figure 6.4: Angle of Repose of Dry Sand
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Exercise and Questions Visual Estimation and Grain Roundness Question #1.
Come up to the front of the lab room and have a look at the sand under the microscope to gain an appreciation for its shape. Which of the following best represents the overall shape of the individual sand grains and make a record of this in the appropriate table (2 marks) Fill in the MC question on Brightspace. Indicate which letter best represents the overall shape of the individual sand grains
:
a. Very angular b. Angular c. Sub-angular d. Sub-rounded e. Rounded f. Well-rou
nded Formulating Hypotheses Consider each of the experiments that you are to do and jot down how you expect the average slope angle to vary between the two different experiments How will the angle of repose (as measured by forming cones) compare with the angle of internal friction (as measured by the tilting boards)? Note: This is to be handed in before the beginning of the experiment. Be sure to keep a copy of what you said as this forms the basis of one of the questions you have to answer. The rest of the lab (i.e., data, interpretations and conclusions) is due in the usual two-
week’s time.
Don’t forget to make a record of what you say, as you’ll need this to answer one of the questions.
EXPERIMENT #1: The Angle of Repose Object: To determine the angle of repose of dry sand as it is poured to form a cone. Procedure: a)
Pour the material through a funnel, or directly from the container onto the paper to form a conical heap. Your goal is to form a perfectly uniform cone, with the steepest possible sides. b)
Measure the slope angle on the side of the heap at four different points, recording each value. c)
Repeat five times for each material (i.e., construct 5 separate cones), for a total of 20 measurements. d)
Question #2.
Record the group data measured by forming the cones in the appropriate table. Indicate the measured slopes angles. There should be four (4) for each cone. Copy and paste the table from the Student Answer Sheet into the appropriate question space on Brightspace. (5 marks) Do not upload the entire word document.
CONE
MEASURED SLOPE ANGLES (4 for each cone)
1 2 3 4 5 e)
Add your group’s data to the pooled class data table on the lab room computer.
EXPERIMENT #2: The Angle of Internal Friction Object: To determine the angle of internal friction for dry sand on a slope of the same material. Procedure: a)
Place the material on the board such that the sand forms a very thin layer on the surface. (Ideally, the layer should be only one particle thick on the surface, but this, of course, is very difficult to achieve.) b)
Begin to raise the board’s angle. You should notice that at first, individual particles begin to slide over themselves. This indicates that the angle of internal friction is being approached. When the majority of material slides down the surface, stop the board and record the angle from the attached inclinometer. c)
After recording the angle, remove any residual material from the surface. Repeat the experiment 20 times. d)
Question #3.
Record the group data in the appropriate table. Indicate the measured board angles (20 Measurements) Copy and paste the table from the Student Answer Sheet into the appropriate question space on Brightspace. (5 marks) Do not upload the entire word document.
MEASURED BOARD ANGLES (20 MEASUREMENTS) e)
Add your group’s data to the pooled class data table on the lab room computer.
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Data Analysis Whenever experiments such as these are performed, a number of trials are run in an attempt to reduce the variance of the results, i.e., “average out the effect of unrealistic numbers”. As this implies, the data from the series of tests is averaged, or the mean is calculated. This calculated mean will give an indication of the magnitude at which an event will occur, or for example, to what angle a given board must be raised in order to initiate the sliding of the material down its surface. We know, however, that the material did not slide down that surface at that specific angle every time, i.e., there was some spread of the data about the mean value. A measure of this spread about the mean can be expressed as the standard deviation. Data such as this usually resembles a normal, or “bell” curve when plotted. This type of graph indicates that the greatest number of data points occur close to the mean value, with a decreasing number of readings occurring as one moves further away from the mean in either direction. The standard deviation is a measure of the dispersion of the data about the mean value. Figure 6.5 –
Bell Curve example
The higher the standard deviation, the greater the spread of data about the mean and the less consistent the data. A problem arises if the data for the two experiments overlap, as shown below. For example, if the maximum value from one material overlaps the minimum value from the other material, do the two materials have the same angle of repose or internal friction? Consider the case where only one measurement was made for each experiment. It is possible that one might have measured the highest angle from the generally lower values and the lowest angle from the generally higher values thus leading to completely incorrect conclusions. It is possible to statistically compare the results of the two experiments and deal with this problem. The question we are really trying to answer is, “Are the overall mean angles obtained from the two experiments significantly different from each other, or are the different mean values the result of chance alone?” The design of our experiment allows several comparisons to be made, some of which will be done in the lab, and some of which you will do on your own as part of the write-up. The most obvious question is: “Does sand have different angles of repose and internal friction?” That is, we can compare the angle of repose for sand, as measured via the cones, with the angle of internal friction as measured via the tilting boards. Although we’re trying to determine the same parameter in each case, the maximum slope angle that sand can attain, and no matter how it’s done (via cones or tilting boards), it still represents the balance between frictional and gravitational forces, the use of two separate experimental methods may give some insight into the actual processes involved. Although every group is performing the same experiment, via the same procedure, it is possible that as a result of, say, minor differences in technique, different groups may obtain different results. A third
question, therefore, “Are the results from my group comparable with the pooled results?” Although it is possible to test the effect of all of the groups on the outcome of the experiment, we’ll focus on how the data from your group compares to the pooled data from the whole lab. You’ll have to compare the data for each of the two experimental methods. Question #4.
- Plot your own group’s Experiment #1 data as a histogram (bar graph) using a spreadsheet (e.g. Microsoft Excel). When you create the graph, place the measured angles on the horizontal x-axis and the number of occurrences of these measured angles (frequency) on the vertical y-
axis. Also, on the graph, indicate the mean and standard deviation from both the pooled results and your own data to get some idea of the angles at which the material moved. Save your graph image as a PDF, .jpg, or .png file (Please note that Apple Pages, Apple Numbers, Excel spreadsheets, or other files will receive a zero, or grade deduction). Once you have created the image file, upload the file to Brightspace. (5 marks) Do not upload the entire word document.
NOTE: Although it is tempting to use the computer to generate these plots, beware that this is not necessarily as easily done as it first seems, especially using EXCEL! Be sure the computer gives you plots in the proper form! Depending on the software, it’s sometimes easier to just do this by hand! ** If you are handing drawing the graph(s), they must be done on graph paper only (not lined paper or blank paper). Question #5 - Plot your own group’s Experiment #2 data as a histogram (bar graph) using a spreadsheet (e.g. Microsoft Excel). When you are creating the graph, place the measured angles on the horizontal x-
axis and the number of occurrences of these measured angles (frequency) on the vertical y-axis. Also, on the graph, indicate the mean and standard deviation from both the pooled results and your own data to get some idea of the angles at which the material moved. Save your graph image as a PDF, .jpg, or .png file (Please note that Apple Pages, Apple Numbers, Excel spreadsheets, or other files will receive a zero, or grade deduction). Once you have created the image file, upload the file to Brightspace. (5 marks) Do not upload the entire word document.
NOTE: that the histograms plotted from your own group’s data will likely not resemble the ideal bell curve more closely approximated by the pooled data, as you are using too few data points.
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Angle of Internal Friction
Frequency
44
42
40
38
36
34
32
30
6
5
4
3
2
1
0
Angle of Internal Friction
Our Group:
Mean = 39.400
Std. Dev. = 1.984
Pooled Data:
Mean = 38.125
Std. Dev. = 2.625
Summary Calculations Question #6
. With the aid of software capable of performing statistical calculations (e.g. Microsoft Excel), compute the mean slope value for the 20 measurements, note the maximum and minimum values, and determine the standard deviation of both your data set and the pooled results for the whole lab. Make a record of this in the appropriate table. (see Class Data folder in the Lab 6 folder on Brightspace). These data will facilitate the statistical analysis of the results. Complete the table in the word document handout and then copy/paste it into the text box in Brightspace. (8 marks) Do not upload the entire word document. EXPT #1 Angle of Repose EXPT #2 Angle of Internal Friction Your Group Pooled Results Your Group Pooled Results Maximum Value Minimum Value Mean Value Standard Deviation
WRITE-UP AND CONCLUSIONS Use the data analysis in describing and explaining any irregularities you may have found by means of the following questions. When discussing any differences in the mean angles obtained from the two experiments, be sure to make reference to the means and standard deviations
. Question #7.
Explain why your results from the pooled data agree or disagree with your hypotheses made before the start of the experiments. Explain how the reasoning you used to formulate your hypotheses was either correct or in error, now that you’ve performed the experiment and have a feel for how the material behaved. Hint, the pooled mean values for each experiment will be the basis for your validation. Write out your answer on the Student Answer Sheet and then copy/paste it into the answer box on Brightspace. (5 marks) Do not upload the entire word document.
Question #8
. If the sand attained different angles of repose and internal friction, propose reasons why this may have occurred, based on your observations and the pooled data of the two experiments. Be sure to consider the results of both experiments. What reasons could account for any differences? It is very tempting to attribute any differences to ‘human error’ but this is usually negligible. It is also tempting to claim you don’t have enough data, but this is precisely why you’re using the pooled results which include in the neighbourhood of 100 data points. Hint, use the pooled means to answer the question. The pooled standard deviation values will indicate how reliable your results may be. Min and max values are not reliable indicators to use. Write out your answer on the Student Answer Sheet and then copy/paste it into the answer box on Brightspace.
(5 marks) Do not upload the entire word document.
Question #9
. Compare your own group’s
data with the pooled results. Propose suggestions as to why differences might result between your results and those of the group as a whole may have occurred. (5 marks) Hint, your answer should include at a minimum, a comparison of means and standard deviations. Min and Max values may also be helpful. Write out your answer on the Student Answer Sheet and then copy/paste it into the answer box on Brightspace. Do not upload the entire word document. Question #10.
The experiments were conducted under idealized conditions. What additional experimental factors would you include to improve this experiment as a model of processes occurring in the real world? Note, this is not an opportunity to suggest ways to make the laboratory experiment function better. Rather, it is an opportunity to suggest ways to make the experiment address some of the ‘real world’ factors that might make our experiments more reflective of conditions we might find in the field. Write out your answer on the Student Answer Sheet and then copy/paste it into the answer box on Brightspace. (5 marks) Do not upload the entire word document.
Please note that all answers will be submitted through the class page on the school LMS. Submitting your Assignment •
It is recommended you write out your answers in a word document (e.g. Student Answer Sheet provided) or similar text document prior to filling in the questions on Brightspace. In this manner you will have a backup in case of a system error (e.g. WIFI lost, etc.), you will be able to proofread your work before submission and you can copy and paste your answer into the system to save time. •
With all your answers complete •
Sign into Brightspace •
Under the ‘Quizzes’ tab on the Upper menu, select “Lab Assignment #5” •
Once in the Quiz page, write out your answer or copy and paste the answers you have pre-written to answer the correct questions. •
DO NOT upload the student answer sheet (word doc) –
all answers must be input manually or copied/pasted into the appropriate space unless otherwise indicated. •
When you are asked to upload an image file (PDF or jpg)
, under ‘browse’, navigate to where you have saved the file on your hard drive (e.g. Desktop) and select the file you want to upload to Brightspace.
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REFERENCES Albert, R., I. Albert, D. Hornbaker, P. Schiffer, and A-L Baraba´si (1997) Maximum angle of stability in wet and dry spherical granular media. Physical Review E, V. 56, # 6, pp. R6271- R6274. Bruce, I.G., D.M. Cruden and T.M. Eaton (1989) Use of a Tilting Table to Determine the Basic Friction Angle of Hard Rock Samples. Canadian Geotechnical Journal, V.26 pp 474-479. Burkalow, A. von, (1945) Angle of Repose and Sliding Friction. Bulletin, Geological Society of America, V.56, pp. 669-708. Courrech du Pont, S., P. Gondret, B. Perrin and M. Rabaud (2003) Wall effects on granular heap stability. Europhysics Lettters, V.61, #4, pp. 492
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498 Frayssea, N., H. Thomé, and L. Petit (1999) Humidity effects on the stability of a sandpile. European Physical Journal, B 11, 615-619. Halsey, T.C. and Alex J. Levine (1998) How Sandcastles Fall. Physical Review Letters, V. 80, # 14, pp. 3141-3144. Kestenbaum, D. (1997) Sand Castles and Cocktail Nuts. New Scientist, V.154, No. 2083, pp.24-28. Mehta, A and G. Barker (1991) The Self-Organizing Sand Pile. New Scientist, V.15, pp. 40-43. Powers, M.C. (1953) A new roundness scale for sedimentary particles. Journal of Sedimentary Petrology, V.23, pp.117-119. Ramana, Y.V. and B.S. Gogte (1989) Dependence of Coefficient of Sliding Friction in Rocks on Lithology and Mineral Characteristics. Engineering Geology, V.26, pp. 271-279. Selby, M.J. (1985) Earth’s Changing Surface, Chapter 6, Behaviour, Strength, and Resistance of Rock, Soil and Water. Oxford University Press. Tegzes,P., T. Vicsek, and P. Schiffer (2002) Avalanche Dynamics in Wet Granular Materials Physical Review Letters V. 89, #9, pp. 094301-1- 094301-4. Young, A. (1961) Characteristic and Limiting Slope Angles. Zeitschrift für Geomorphologie, V. 5, #2, pp. 126-131.