Slippery Slopes and the Angle of Repose
Have you ever seen an avalanche or landslide roll down a hill? Why is it that at one moment, everything seems fine, then suddenly the mountain begins to slump? It has something to do with how the earth or snow is piled up on the mountain. Generally, granular materials such as snow or earth pile up relatively well. However, if the slope angle gets too steep, the materials will start to slide down the slope. This critical slope angle, also called the angle of repose, is different for different materials. In this activity, you will create your own little avalanches and determine the angle of repose for different materials along the way!
- Adult helper
- Plastic cup, 16 oz.
- Printer pager
- Baking dish
- Tape measure
- Lentils (2 cups)
- Powdered sugar (2 cups)
- Table salt (2 cups)
- Rice (2 cups)
- Scientific calculator
- Optional: Protractor
- With the help of an adult, cut a small hole in the bottom of the plastic cup. Its diameter should be about 2 cm.
- Place a sheet of printer paper into the baking dish and label it with the material that you want to test.
- Pour the first material into the prepared plastic cup. Cover the hole with your hand or fingers and fill the cup at least halfway.
- Hold the cup on top of the printer paper at its center. Then, remove your hand or fingers to release the material inside the cup.Watch the material as it piles up in the baking dish. What do you notice?
- As the material pile grows, hold your cup higher so all the material can fall onto the paper. Depending on the material, you might need to tap the cup a little to help it all come out.Observe the slope angle of your pile as it grows. Does it change over time? How does the material pile and the slope angle look at the end?
- With a pen, carefully draw along the circumference of the material pile. Be careful not to disturb the pile too much.
- Next, take the ruler and measure the height (h) of the material pile in centimeters [cm]. You can very carefully slide the ruler into the pile to measure its height at its peak. If this method disturbs the pile too much, you can also hold the ruler next to the pile and carefully extend a tape measure from the top of the pile to the ruler. The height of the pile is where the tape measure and ruler intersect. Write down the height of the pile on the sheet of paper next to the pile.
- Using your drawn circle and the measured height, calculate the angle of repose for this material. The equation for calculating the angle of repose is tan-1(h/r). Don' worry if the equation looks scary—you will determine each number step by step, and the rest is done by the calculator!
- First, remove the material from the sheet of paper. Using the ruler, measure two different diameters (d) of the drawn circle in centimeters [cm]. To do this, draw two lines from one random edge of the circle through the center of the circle to the opposite edge of the circle. The length of each line will give you the diameter of the circle. Write down both numbers.Are both numbers very different? What does this tell you about the shape of the circle?
- Next, calculate the average diameter of your circle by adding both measured diameters and dividing the result by two. From this you can calculate the radius (r) of your circle by dividing the average diameter by two again.
- Use the calculator to divide the measured height [in cm] by the calculated radius [in cm]. Write down the result to one decimal point.
- Now, the only step left is to enter this number into the calculator and hit the inverse tan key (or tan -1). This will give you the angle of repose. Write it down on your sheet of paper.
- Repeat steps 1–11 with all the other materials.How do all the different materials compare? Does the shape and size of each material pile differ? Which material has the lowest or highest angle of repose? Did you expect these results?
How did your piles look? Each of your materials should have formed a nice conical pile. The circumference of each pile should have been close to a symmetric circle, which means that the two measured diameters should be relatively similar. However, the height of the piles and the size of the measured circles should have changed depending on the materials you tested. While lentils and rice form large circles, powdered sugar results in a very small circle. Conversely, the heights of the lentil and rice piles should have been significantly smaller than the height of the powdered sugar pile.
Based on these numbers, you probably found that lentils and rice have a small angle of repose (around 25-35°), while the powdered sugar has a relatively high angle of repose (>40°). The calculated angles of salt and flour should be somewhere in between those. This variation is due to the different sizes and shapes of the material particles. Generally, increasing particle size will decrease the angle of repose. This is why the large lentil and rice particles have a much lower angle of repose than that of the fine-grained powdered sugar. Also, particles that are irregularly shaped hold together much better than particles that are very round and easily roll over each other.
If you pour a granular material on a plane surface, it will form a conical pile. If you add more of the material, the pile will grow. However, at some point the slope angle of the pile will always stay the same. This is because as the pile grows and its slope reaches a certain angle, some material will slide down the pile. This so-called angle of repose is the steepest angle at which a material can be heaped without sliding down. But why would the material slide?
The reason is gravity. The gravitational force acting on the material on the slope can be split into two different components. One component, the normal force, pulls the material into the slope in a direction perpendicular to the slope surface. The normal force pulls inward on the grains on the slope, which actually helps hold the grains together and prevents the material from sliding downward. Depending on the type and shape of material, frictional forces between the grains may also hold them together. As a result, grains of irregular shape that have the ability to interlock tend to have a higher angle of repose. The second gravitational component is the shear force, which pulls the grains down the slope in a direction parallel to the slope's surface. The steeper the slope, the higher the shear force will be. At some point, the shear force will overcome the normal force of gravity. This is usually the moment the materials start sliding down the slope and the angle of repose is reached.
This might sound like a very theoretical concept. However, there are plenty of situations where grains such as corn, flour, or gravel need to be piled up. In these situations, knowledge of their angle of repose can be very helpful in figuring out the proper dimensions of a storage silo, or to design the right-sized conveyor belt to transport them. In addition, the angle of repose is used to assess whether a slope is going to collapse. This helps geologists or mountaineers know the risks of avalanches ahead of time! There are several ways to measure the angle of repose of a specific material. One, which you will be doing in this activity, involves measuring the height and radius of a pile formed by a material, then using these numbers to calculate the angle of repose.
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For Further Exploration
- Test some more materials. Any granular material will work. Other possibilities to test are sand, coffee beans, gravel, cornstarch, etc. How do these materials compare with the ones you have tested?
- Use a protractor to measure the angle of repose directly from the pile you created with each material. How well does your measured angle match up with your calculated angle of repose?