How steep can a sand pile be?
Tonight's idea
Pour dry sand slowly onto a table. It makes a cone. Keep pouring and the cone grows, but its sides keep about the same slope. If a side gets steeper, grains slide down in a small avalanche until the slope is back where it was.
The steepest slope a loose pile can hold is its angle of repose. For dry sand it is about 30 to 35 degrees: nearer 30 for smooth, round grains and steeper for rough, angular ones.
So a small pile and a big pile of the same sand are the same shape at different sizes. Double the width and the height doubles too.
Gravel, grain and dry soil each have their own angle. This is a simple model: a real pile wobbles a degree or two around its angle.
Watch one: measure a pile, then predict a bigger one
Take the small pile in the picture: 20 cm across and 6.5 cm high. The radius is half the width: 10 cm. From the edge to the middle, the slope climbs 6.5 cm over 10 cm.
\[\text{steepness} = \text{height} \div \text{radius} = 6.5 \div 10 = 0.65\]
The slope climbs 0.65 cm for every 1 cm you move in. A scale drawing turns this into an angle of about 33 degrees (exercise 1).
Now predict. A pile of the same sand is 50 cm across, so its radius is 25 cm. The model says its steepness is still 0.65, so its height is \(25 \times 0.65 = 16.25\) cm. Call it about 16 cm: the angle is only roughly the same.
Your turn
- You need a ruler, a protractor and spare paper. Draw a line 10 cm long. At its right end draw a line straight up, 6.5 cm long. Join its top to the left end. This is half of the small pile, seen from the side, at full size. Measure the angle at the left end. Is it between 30 and 35 degrees?
- Use the steepness 0.65. (a) A pile of the same sand has radius 1 m. How high is it, in cm? (b) Another pile is 2.6 m high. What is its radius, and how wide is it across?
- Pile A is 20 cm across and pile B is 40 cm across, both of the same sand. (a) How high is B? (b) A cube 2 cm along each edge is built from 1 cm cubes. How many does it take? (c) The same thing happens to any shape when every length doubles. Pile A holds 1 cup of sand (made up). How many cups does B hold? How many would a pile 60 cm across hold?
- Draw a line 10 cm long. At its left end draw a slope at 30 degrees, and at its right end a line straight up to meet it. Measure the height. Do it again for 35 degrees and for 45 degrees. So how high can a dry sand pile 20 cm across be?
- In damp sand, a little water makes tiny bridges between grains that touch, and the bridges pull the grains together. (a) A sandcastle wall stands straight up. What angle does it make with the ground? (b) Why can the steepness 0.65 not predict the shape of a damp pile? (c) What do you think happens to the wall when the sand dries out?
Stretch: keep pouring
Sand trickles onto the top of a pile at a steady rate, one cup every minute (made up). After 1 minute the pile is pile A: 20 cm across and 6.5 cm high. In this model, at what time is it 40 cm across? 60 cm across? How high is it after 64 minutes? Describe how the growth of the pile changes as time goes on.
Answers
- About 33 degrees, which is between 30 and 35. Anything from 32 to 34 is good measuring. So this made-up pile is a believable pile of dry sand.
- (a) \(100 \times 0.65 =\) 65 cm, since 1 m is 100 cm. The radius is 10 times that of the small pile, so the height is 10 times 6.5 cm. (b) Height is radius times 0.65, so radius is height divided by 0.65: \(2.6 \div 0.65 =\) 4 m. Check: \(4 \times 0.65 = 2.6\). The pile is 8 m across.
- (a) B has radius 20 cm, so its height is \(20 \times 0.65 =\) 13 cm, double the height of A. (b) \(2 \times 2 \times 2 =\) 8 small cubes. (c) B is A with every length doubled, so it holds 8 cups. A pile 60 cm across is A with every length tripled, so it holds \(3 \times 3 \times 3 =\) 27 cups. Twice as wide is far more than twice the sand.
- About 5.8 cm at 30 degrees and about 7.0 cm at 35 degrees. Being 2 mm out is fine: one degree on the protractor changes the height by 2 or 3 mm. At 45 degrees the height is 10 cm, the same as the base, because the triangle's other corner is also 45 degrees and so two of its sides match. A dry sand pile 20 cm across can be at most about 6 to 7 cm high, depending on the sand. It would need a 45 degree slope to be as high as its radius, and dry sand does not hold one.
- (a) 90 degrees. (b) The steepness 0.65 comes from loose grains that only rest on each other, so they slide when the slope is too steep. In damp sand the water bridges hold grains in place, so the sand can keep a much steeper shape, and the shape depends on how it was packed. There is no single angle. (c) As the water dries up the bridges disappear. The grains are loose again and the wall crumbles into a slope near the dry angle of repose.
Stretch. The pile 40 cm across holds 8 cups (exercise 3), so it takes 8 minutes. The pile 60 cm across holds 27 cups: 27 minutes. After 64 minutes it holds 64 cups, and \(4 \times 4 \times 4 = 64\), so every length is 4 times that of pile A: 80 cm across and \(4 \times 6.5 =\) 26 cm high. The pile keeps its shape, but it grows more and more slowly. The first 6.5 cm of height took 1 minute, the next 6.5 cm took 7 minutes, the next took 19 and the one after that took 37. Each new layer has to cover a bigger cone. Up close a real pile does not grow smoothly: the top steepens a little, then an avalanche carries sand down the side, again and again.