Investigating the Impact of Roof Angles on Snow Falling
Mr. Doner
SL Physics 2017/18
March 7 2018
Table of Contents
Introduction 2
Research Question 2
Exploration & Hypothesis 2
How the experiment was conducted 3
Materials 4
Procedure 5
Data 5
Explanations 9
Conclusion 10
Suggestions for Improvement 11
Works Cited 12
Introduction
A while back, my parents booked a new house but a few months later, they realised that the incline of the roof in a certain area did not look very aesthetically appealing. They went to the builder and asked if the slope of the roof could be decreased due to the lack of aesthetic appeal. However, the builder declined, citing government regulations. According to those regulations, a certain angle is required to make sure that snow, amongst other types of precipitations, falls off the roof fast enough that it can’t pool on the rooftop. Although my parents eventually agreed, I questioned why the roofs in other areas did not have to be as steep as ours. This scenario was the inspiration for my physics exploration.
Research Question
After realising that I could explore any topic that interested me for my investigation, I narrowed my exploration down and was left with the following research question: What is the minimum incline needed on a roof for snow to slide off and thus prevent the pooling of snow?
Exploration & Hypothesis
Personally, I believe that the minimum angle that’s required could vary significantly, depending on a few factors: the consistency of the snow, how fast and how much is falling, and the friction between the snow and the rooftop. Considering that different parts of the world use different materials for their rooftops, varying anywhere from wood to asphalt to rubber, it would be quite difficult to find an angle that would be generally applicable and acceptable world-wide.
After analysing and reflecting on how my experiment will work, I plan on using sand in the place of snow and changing its consistency to imitate actual snow with the help of water (refer to the next section). My hypothesis is that the minimum angle will be above 20° for the completely dry sand, but closer to 40° on average for the different consistencies of wet sand. This is because the water added will bind the sand particles together, which will cause all the individual fine particles to cluster together and create one large mass that will be forced to move as an single body instead of as individual bodies.
How the experiment was conducted
The dependent variable in my experiment is the angle created by the different consistencies of wet sand. The independent variable in my experiment is the consistency of the wet sand.
The goal of this experiment is to calculate the minimum angle that snow creates on a rooftop. However, it is not realistic to use snow due to weather conditions, so sand was used to imitate the snow. In addition to this, a wood plank was used to mimic the rooftop, since a rooftop would not be applicable in a classroom setting. To measure the angle of the rooftop at different consistencies of snow, different amounts of water were used.
Keeping in mind that snow has different consistencies, it would be difficult to experiment what the minimum angle would be using only one consistency sand. For this reason, I plan on finding the minimum angle created with 20g of dry sand, and then seeing what would happen to the angle if 0.05g to 0.25g of water were mixed in.
Below is a diagram that shows how the experiment was conducted and the sides that would be measured and recorded are labelled.
All recorded measurements were made using a meter stick or an electronic scale.
An undesirable effect that needed to be minimised was the impact of dropping the sand from an inconsistent height. For this reason, I felt it was necessary to drop the sand from the height of 65 cm, to make sure all the sand had an equal force and velocity on the wood plank when it fell.
Materials
I used a wood plank of length 59.4cm and width 20.1 cm. It was unwaxed and was slightly rough as to make it as similar as possible to a wooden roof shingle, which are typically rough.
I used 20 g of regular park sand consistently throughout the experiment.
I used a dropper that dispensed approximately 0.05g for every drop.
The water used was regular tap water that was put into the dropper to be deposited into the sand.
I used regular textbooks (not a specific type) to lean the wood plank against while measuring the opposite and adjacent angles. A textbook was also used as a stopper to prevent the end of the plank from sliding around.
To make the measurements, a metre stick was used.
To weigh the sand and water, an electronic scale was also used.
Beakers of 80 mL were used to hold the wet sand mixtures.
Procedure
Measure beaker mass and record
Measure 20g of sand and record.
Drop the sand onto one extreme of the wood plank, but from a distance of _____ cm from the adjacent side indicated in the diagram above.
Slowly lift the wood plank from the side the sand was poured onto. Measure and record the opposite and adjacent sides when the sand begins to fall over itself.
Continue to lift the wood plank until the sand begins to slide down the ramp. Measure and record the opposite and adjacent sides once again.
Repeat steps 1-5 five times for each consistency of wet sand. Add an additional drop of water for each consistency (0 drops, 2 drops, 3 drops, 4 drops, 5 drops) in step 2. Make sure to mix the sand and water, then record the combined mass.
Data
Consistent Variables:
Beaker Mass: 50.84g
Amount of Sand: 20g,0.3g
1 Drop of Water 0.05g
How the Processed Data is Calculated:
To calculate the mass of the sand & water mixtures, the mass of the empty beaker was subtracted from the mass of the beaker containing both the sand and water. For example, if the sand+water+beaker is 70.85g and the beaker is 50.84g, then the sand+water mass is equal to 70.85g – 50.84 = 20.01g.
To calculate the angles, I calculated the inverse tangent of the opposite over the adjacent. For example, if the opposite is 30cm and the adjacent is 47.5, then tan-1 (3047.5) 32.3°.
Averages were calculated by taking the mean of the data set in the column. For example,
(26.78 + 26.78 + 28.03 + 21.21 + 26.12 ) = 136.92/5 = 27.384°.
Uncertainties were calculated using the (max – min)2 method. For example, (28.03 – 26.12)2 = 1.545.
The reason I looked into the angle the sand starts sliding over itself in addition to the angle that the sand starts to slide down the ramp is because in typical winters, there is a good chance that there might be a thin layer of snow on a roof that would stay there permanently. By “permanently”, I mean in the sense that it would not get a chance to melt before more snow falls on top of it. This means that anything that builds on top of the bottom layer would have to slide off first, going over the layer below it.
Raw Data is on the left side and Processed Data is on the right side of the thick line.
Table 1: 0 Drops of Water
Trial
Sand, Beaker, Water Mass/g
Sand Slides over itself/cm
Sand Falls Down Ramp/cm
Sand & Water Mass
Angle: Sand falls over itself
Angle: sand falls down the ramp
Opp.
Adj.
Opp.
Adj.
1
70.82
26.0
51.5
30.0
48.0
19.98
26.78
32.00
2
70.83
26.0
51.5
30.9
48.3
19.99
26.78
32.61
3
70.85
27.0
50.7
29.0
47.0
20.01
28.03
31.67
4
70.83
27.4
49.0
31.2
49
19.99
29.21
32.49
5
70.83
25.5
52.0
32.1
49.1
19.99
26.12
33.15
Ave
19.992
27.38
32.38
Unc
0.015
1.55
0.74
1 drop of water was not substantial enough to be able to fully mix with the sand; it clumped up in little piles. This resulted in data that did not show realistic results since the majority of the sand was still completely dry. For this reason, I could not collect sufficient data to account for what would happen if there were 1 drop of water for every 20g of sand.
Table 2: 2 Drops of Water
Trial
Sand, Beaker, Water Mass/g
Sand Slides over itself/cm
Sand Falls Down Ramp/cm
Sand & Water Mass
Angle: Sand falls over itself
Angle: sand falls down the ramp
Opp.
Adj.
Opp.
Adj.
1
70.93
29.0
48.5
33.0
45.0
20.09
30.88
36.25
2
70.97
30.0
47.6
34.0
44.0
20.13
32.22
37.69
3
70.94
30.0
47.5
37.0
41.0
20.10
32.28
42.06
4
70.95
29.0
48.4
34.0
44.0
20.11
30.93
37.69
5
70.94
30.5
47.0
33.0
45.0
20.10
32.98
36.25
Ave
20.106
31.86
37.59
Unc
0.02
1.05
2.91
Table 3: 3 Drops of Water
Trial
Sand, Beaker, Water Mass/g
Sand Slides over itself/cm
Sand Falls Down Ramp/cm
Sand & Water Mass
Angle: Sand falls over itself
Angle: sand falls down the ramp
Opp.
Adj.
Opp.
Adj.
1
70.99
34.6
44.1
38.21
40.3
20.15
38.12
43.47
2
71.0
34.5
44.0
37.9
41.3
20.16
38.10
42.54
3
71.0
34.5
44
38.0
40.5
20.16
39.10
43.18
4
71.01
34.0
43.5
37.5
41.0
20.17
38.01
42.45
5
70.99
33.5
43.0
38.1
40.5
20.15
37.92
43.25
Ave
20.158
38.25
42.98
Unc
0.01
0.59
0.51
Table 4: 4 Drops of Water
Trial
Sand, Beaker, Water Mass/g
Sand Slides over itself/cm
Sand Falls Down Ramp/cm
Sand & Water Mass
Angle: Sand falls over itself
Angle: sand falls down the ramp
Opp.
Adj
Opp.
Adj.
1
71.09
38.0
40.5
41.5
37
20.25
43.18
47.59
2
71.05
37.5
41.0
42.1
36.6
20.21
42.45
48.98
3
71.08
37.8
41.2
42.5
36
20.24
42.54
49.73
4
71.03
37.1
40.9
42.0
36.7
20.19
42.21
48.85
5
71.05
36.9
40.7
41.9
37.4
20.21
42.19
48.24
Ave.
20.22
42.51
48.68
Unc.
0.03
0.49
1.07
Table 5: 5 Drops of Water
Trial
Sand, Beaker, Water Mass/g
Sand Slides over itself/cm
Sand Falls Down Ramp/cm
Sand & Water Mass
Angle: Sand falls over itself
Angle: sand falls down the ramp
Opp.
Adj.
Opp.
Adj.
1
71.10
44.5
33.0
53.1
27.9
20.26
53.44
62.28
2
71.10
44.0
33.5
53.8
27.2
20.26
52.72
63.18
3
71.09
43.8
33.2
53.5
27.0
20.25
52.84
63.22
4
71.10
45.0
32.5
54
26.5
20.26
54.16
63.86
5
71.11
44.6
32.9
54.1
26.9
20.27
53.58
63.56
Ave
20.26
53.35
63.22
Unc
0.01
0.72
0.79
Graph Variables & Graph
X: Drops of water
Y1: Angle when sand falls over itself
Y1 Uncertainties
Y2: Angle when sand falls down ramp
Y2 Uncertainties
0
27.38
1.55
35.56
1.08
2
31.86
1.05
37.59
2.91
3
38.25
0.59
42.98
0.51
4
42.51
0.49
48.68
1.07
5
53.35
0.72
53.35
63.22
Explanations
After analysing the data and resulting graph above, a positive correlation between the angle created and the wetness of the sand can be seen. This means that the relationship between the water and sand is that as more water was added the angle for the roof became steeper. If wetter sand requires a steeper angle before it starts moving, a possible cause for this could be friction. This led me to do some research online, and I found a website that gave information on different coefficients friction for snow. Although the website entry is short, it gives coefficients of friction for snow on a variety of surfaces, and the values ranged from 0.01 to 1.76, depending on the surface. The large range in the values prompted me to find the coefficient of friction for my own data at the different angles to see if they matched up with those given for wood and snow.
To calculate the coefficient of friction, I started off by finding the frictional force using mgsinθ, and then I found the normal force through mgcosθ and then divided them. For example, if m = 19.99, g = 9.8 and θ = 27.38, then the frictional force would be mgsinθ = (19.99)(9.8)(sin 27.38°) 90.09 and the normal force would be mgcosθ = (19.99)(9.8)(cos 27.38)173.96. To find the coefficient of friction, μ, you would let μ = 90.09 173.96 0.518. My data is as follows:
Drops of Water
for When the Sand Falls Over Itself
for When the Sand Slides Down the Ramp
0
0.518
0.715
2
0.621
0.769
3
0.788
0.932
4
0.917
1.137
5
1.344
1.981
The trend shown is that increasing the amount of water increases the coefficient of static friction. This goes back to what I mentioned in hypothesis was the hypothesis right? State that , about how the water will bind the sand particles together, thus creating one large mass. If the sand particles bind together to create one large mass, it makes it more difficult for the mass to move, thus increasing the amount of friction present. At this point, if you wanted to make the mass move, you would have to increase the angle of the rooftop. It is now possible to see that there is definitely a correlation between these three factors; in simpler terms, the wetness of the sand increases the friction and if someone were to make the mass of sand move, you would need to make the angle steeper.
As mentioned above, based off of the website I looked at, the coefficient of friction on different snow types ranged from 0.53 to 1.76. The coefficients of friction that I calculated fall into the range of approximately 0.52 to 1.98. Given that my range is quite similar, I think it is fair to say that using sand as a substitute for snow gave fairly accurate results that can be deemed as reasonably relevant. Looking further into the study, I found that drier snow (more frozen) had lower coefficients of friction than the wetter sand (less frozen), similar to what I found in my experiment with the dry sand having a smaller coefficient of friction than the wet sand, which is once again applicable to how a mass with a higher coefficient will require a larger force to counter the friction force, resulting in a steeper incline.
Conclusion
Although there is definitely a positive correlation between the wetness of the sand and the steepness of the angle created, my exponential line of best fit does not pass through all the error bars, leading to three conclusions: it is necessary to collect more data to get a better range of the necessary angle; that the steepness of the minimum angle varies exponentially; and it is quite possible that my data is not as precise as my uncertainties led me to believe. If my data were more precise, the trend the data seems to be following would be more clear. Since my exponential line of best fit does not pass through all my data points, there are definitely some inaccuracies that would need to be accounted for in improved versions of this experiment.
My hypothesis was that a steeper angle would be required for the wetter sand to fall down the ramp because the mass would stick to itself. My hypothesis did turn out to be true, but I now know the reason behind its truth: friction plays a huge role! I also now know that the minimum angle needed can vary a lot more significantly than I thought, depending on the snow. Personally, it seems reasonable for a generally accepted angle to be on the steeper side, just for peace of mind in the sense that it would help prevent worry about snow pooling on rooftops simply because it didn’t fall off. Ultimately, although the very minimum angle I found for the dry sand to start sliding down the ramp was 32.38°, I realise now that since most snow would be less frozen (wetter) than I’m assuming it to be, so it would make sense for an ideal angle to fall somewhere between 48° to 63°.
Given that the information I found is very consistent with the information given in other studies, It certainly suggests that the conclusions I made and the new information I found is quite relevant.
Suggestions for Improvement
After reviewing my experiment and reflecting on it, I found that there are a near equal number of significant strengths and weaknesses. One of the most significant advantages to the way I conducted my experiment is that I take different consistencies of snow into account; if I had used only one consistency of wet sand (or dry sand), I would not have been able to get such a realistic range of data values. Another strength I believe my exploration has is that if someone wanted to conduct this experiment on a larger scale, they have a general idea of what the minimum angle would be and see if the angle found here is still relevant. Furthermore, after analysing how the coefficient of friction plays a role in how the angle is affected, it helps give a foundation for those who will conduct this experiment in the future should they want to experiment with multiple roofing materials.
In regards to limitations, most of them can be improved upon, as I will discuss below. Although I took different consistencies of snow into account, actual rooftops face many different types of precipitation, like hail and sleet, and my investigation does not reflect that. Another limitation is that the scale on which this exploration was done is so small compared to how big actual rooftops are. This means that there is no guarantee that the minimum angle is even close to the one I found; the likelihood of real life snow weighing more that 20g and the rooftop being more than ___ cm by ___cm is very high. Moreover, only one type of surface was used even though there are multiple different types of roofing that one could have on their house.
To improve this exploration, there are a number of things that could be done. One of the main improvements would be to conduct this experiment with different types of precipitation because a rooftop would have to battle multiple conditions, so taking into account what would happen to other types of precipitation is important. Different types of precipitation like hail or sleet could be modelled using materials like very small pebbles.
Another improvement would be to use different surfaces. Wood planks and roof shingles have different coefficients of friction and are made of different materials. Doing this would give and idea of whether the minimum angle found in this experiment is still relevant to the different types of roofs one could have. Perhaps, if roof shingles are accessible, the experiment could be conducted using those to get more realistic results.
Works Cited
“17 Types of Roof Shingles [The Complete Guide].” 918 Construction, 19 Jan. 2018, www.918construction.com/types-of-roof-shingles/.
Elert, Glenn. “Coefficients of Friction for Snow.” Coefficients of Friction for Snow – The Physics Factbook, hypertextbook.com/facts/2007/TabraizRasul.shtml.
Gleason, J. Andrew. “Preliminary results of snow surface friction coefficient measurements.” Montana State University Digital Initiatives and Digital Collections, 2002,arc.lib.montana.edu/snow-science/objects/issw-2002-523-527.pdf.