**Key concepts**

Physics

Gravity

Forces

Angle of repose

**Introduction**

Have you ever seen video footage of an avalanche or landslide rolling down a hill? Why is it that at one moment everything seems fine then suddenly the mountain begins to slump? This movement has something to do with how the snow or soil is piled up on the mountain. Granular materials such as snow or soil generally pile up relatively well. If the slope angle gets too steep, however, 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 small avalanches and determine the angle of repose for different materials along the way!

**Background**

If you pour a granular material on a flat surface, it will form a conical pile. If you add more of the material, the pile will grow. At some point, however, 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 is the angle of repose and 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, 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 might 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. There are plenty of situations, however, in which 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-size conveyor belt to transport them. The angle of repose is also used to assess whether a mountain 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.

**Materials**

- Adult helper
- Scissors
- Disposable plastic cup, 16 ounces
- Printer paper
- Baking dish
- Pen
- Ruler
- Tape measure
- Table salt (at least one cup)
- Rice (at least one cup)
- Powdered sugar (at least one cup)
- Scientific calculator or app with scientific calculator functions
- Additional granular materials such as lentils, flour, etcetera
- Protractor (optional)

**Preparation**

- With the help of an adult, cut a small hole in the bottom of the plastic cup. Its diameter should be about two centimeters.

**Procedure**

- Place a sheet of printer paper into the baking dish and label it with the material that you want to test.
- Covering the cup’s hole with your hand or fingers, fill the cup at least halfway with your first material.
- Hold the cup close to the top of the printer paper at its center. Then remove your hand or fingers to release the material inside the cup.
*What do you notice when the material falls on the paper?* - 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 the slope angle change over time? How do the material pile and 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.
*How big or small is your circle?* - With the ruler measure the height (h) of the material pile in centimeters. 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.
*How tall did your material pile get?* - 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't worry if the equation looks complicated—you will determine each number step by step, and the rest is done by the calculator! - Remove the material from the sheet of paper. Using the ruler, measure two different diameters (d) of the drawn circle in centimeters. To do this, draw two lines from one random edge of the circle through its center to its opposite edge. The length of each line will give you the circle’s diameter. Write down both numbers.
*Are the numbers very different? What does this tell you about the shape of the circle?* - 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 centimeters) by the calculated radius (in centimeters). 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 tangent key (or tan-1). This will give you the angle of repose. Write it down on your sheet of paper.
- Repeat these steps with all the other materials.
*How 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?* **Extra:**Test some more materials. Any granular material will work. Other possibilities to test are sand, coffee beans, gravel, cornstarch, etcetera.*How do these materials compare with the ones you have tested?***Extra:**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?*

**Observations and 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 the two measured diameters should be relatively similar. The heights of the piles and the sizes of the measured circles, however, should have changed depending on the materials you tested. The rice probably formed large circles whereas powdered sugar likely resulted in a very small circle. Conversely, the height of the rice pile should have been significantly lower than that of the powdered sugar.

Based on these numbers you probably found rice has a small angle of repose (around 25 to 30 degrees) whereas the powdered sugar has a relatively high angle (greater than 40 degrees). The calculated angle of salt should be somewhere in between those. This variation is due to the different sizes and shapes of the material particles. Increasing particle size will generally decrease the angle of repose. This is why the large rice particles have a much lower angle than that of the fine-grained powdered sugar. Additionally, particles that are irregularly shaped hold together much better than particles that are very round and easily roll over each other.

**Cleanup**

If you used clean materials, you can reuse rice, salt and powdered sugar. Clean your work area, and wash your hands.

**More to explore**

Factors That Control Slope Stability, from *Physical Geology*

Landslides: What Causes Rocks to Slide Down a Slope?, from Science Buddies

Sliding Science: How Are Landslides Caused?, from *Scientific American*

Landslide and Other Gravity Movements, from Science Clarified

STEM Activities for Kids, from Science Buddies

*This activity brought to you in partnership with Science Buddies*