July 12, 2015


An interesting consideration is the angle that a pile of stuff takes on. A pile could be of any substance that is powder-like on a proper scale, that is a pile of sand, dirt, trash, etc. Here we assume that stuff is dropped on top of the pile and then consider the energetics of the following events to see where this stuff ends up.

Clearly the initial gravitational energy of the dropped stuff needs to be dissipated. If this energy is small, friction against existing stuff in the pile will tend to stop the added stuff near the top of the pile. If the gravitational energy is large enough, there will be preference for the stuff to fall down to lower levels of the pile despite the friction from the other stuff. Now consider the amount of gravitational energy required to overcome friction forces. At an edge of the pile, an object may either stay in its place held by friction forces or move down and thus gain gravitational energy but lose energy to the friction forces. Here we assume friction forces are independent of velocity, where in reality friction will likely decrease as the object gains speed and rotational inertia (which is why avalanches are dangerous). As long as friction does not increase with velocity, this analysis will be accurate.

It is straightforward that if the gain in energy from gravity in the fall is exactly countered by friction from the pile the object will continue on its path with the same velocity, and otherwise it will either slow down or speed up. In fact, this should be the condition at the edge of the pile, because if energy gained by falling is greater than friction (steep edges like a column) the pile will tend to disperse, while if the energy required to overcome friction is greater than that gained by falling (flat edges like a puddle) then the stuff dropped on top of the pile will simply add another flat layer that will expand in size until the energies are equal (at which point the stuff will fall down at the edges). So then at the edge of the pile, the two possible energy changes have to be equal. Furthermore, this suggests an interesting pathway in which the pile grows: as more stuff is dropped on top, it will tend to travel towards the bottom of the pile as far as it can at constant velocity (because there is no net energy change at the edges) and then stop as it is trapped in place by friction - other stuff dropped later will stop when it reaches the stuff before it and so on - and thus the pile grows by individual particle-thick layers each of which is geometrically similar to the surface of the pile.

It is important to give more consideration to the friction force. For most materials, static friction force (force exerted before movement begins) is higher than kinetic friction force (force exerted when object already moving). For example, it is difficult to move furniture on carpet from a standstill but once motion is established keeping it in motion is easier. So we can expect that both of these forces play a role in pile formation. If the pile is formed slowly, such as during snowfall, the static friction force dominates, while if the pile is formed quickly by dumping objects on top, the kinetic friction force dominates. Since the static force is higher, the gravitational energy gain at the edge needs to be greater than that for the kinetic case to satisfy the above equality. Greater gravitational energy gain corresponds to steeper pile angles (more column-like). This means that a carefully formed pile will have steeper edges than a thrown-together pile and still be stable.

Let's consider the pile edge angles. The angle of the edge is a representation of both the energy gain due to gravity if a particle were to fall along that angle, and energy loss to friction forces. To satisfy the energy equality the angle must be the same for a particular set of forces, and the angle would be determined by this set of forces. A column-like structure can only be obtained with strongly bound materials such as solid plastic/metal/concrete, while a puddle-like structure is obtained with weakly bound materials like marbles or water. We already determined that two forces are of great importance in friction - static and kinetic - and now point out that the transition is discontinuous, with forces between the static and kinetic being unstable (that is, either the static or the kinetic are in effect but not an in-between force). Thus we expect that two angles will dominate pile edge formation - a steep angle associated with static equilibrium and a less steep angle associated with kinetic equilibrium, and this may be readily observed in real piles.

A real salt pile, taken from https://emilyanddrew.wordpress.com/2008/05/13/salar-de-uyuni-bolivia/imagen-058/. The overall pile angle as well as small sections that are nearly vertical can be seen.

Finally let's consider stability of the pile. The kinetic equilibrium angle is a stable equilibrium - if a particle at this edge falls a bit it will not gain any energy - thus this angle will persist over time. Angles that are less steep than this are also stable over time, because gravitational energy is lost by the particle, but if the pile keeps building up these angles will be unfavorable compared to kinetic equilibrium since stuff will just pile up on top of them until that angle is reached. The static equilibrium angle however is an unstable equilibrium - if a particle at this edge falls a bit the static friction force no longer applies and it gains energy as it falls down a steeper angle than kinetic equilibrium allows. Thus the steeper static equilibrium angles will not persist over time and piles containing such edge angles will tend to collapse and release energy - this is what happens in an avalanche or a landslide. (Update: this angle is known as 'angle of repose')