Physics · Mechanics · Grade 7

Pressure of a solid

Why don't skis sink into snow, while a thin heel punches through it?

The situation

Thick snow has fallen on a field at the foot of the mountains. A person in boots sinks into the snow up to the knees, while a skier glides over the top. A girl in high-heeled shoes punches through the snow with every step. Their masses are almost the same — so why does the snow support them so differently, and what footwear do you need so as not to sink into snow?

Topic

Pressure of a solid. Why a ski does not sink into the snow but a heel does

Key ideas

Pressure p = F/A
Contact area
Compaction of snow

Formulas

p = \frac{F}{S}

Pressure is the force acting perpendicular to a unit area of a surface. $1 Pa = 1 {N/m}^{2}$ .

p = \frac{m g}{2 S_{1}}

A person standing on two feet: the gravitational force is shared between the two soles ( $S_{1}$ — the sole of one foot).

ρ (z) = \frac{ρ_{0} H}{H - z}

When the column of snow under the foot is compressed by $z$ , its density increases (mass is conserved).

σ (ρ) = σ_{m} {(\frac{ρ}{ρ_{m}})}^{4}

In the model, the compressive strength of snow ( $σ_{m} = 4.9$ MPa, $ρ_{m} = 917$ kg/m³ — the density of ice). The foot stops when $σ (z) = p$ .

1 atm = 101 325 Pa \approx 760 mmHg

Non-SI units of pressure: the standard atmosphere, the millimetre of mercury, $1 bar = 10^{5}$ Pa. Converting units ( $1 {cm}^{2} = 10^{- 4} m^{2}$ , $1 kPa = 1000$ Pa) is where calculations most often go wrong.

Theory

What is pressure?

When the same force acts on a small area, the pressure is large: $p = F / S$ . A person's weight $F = m g$ is passed to the snow through the sole of the shoe. If you stand on two feet, each foot carries $m g / 2$ ; if you stand on one foot, the pressure doubles. In the scene the red arrow is the force and the orange outline is the contact area.

How does snow give way? (model)

Fresh snow is ice crystals with air between them. When a foot presses, the column of snow beneath it is compressed and its density rises as $ρ = ρ_{0} H / (H - z)$ , and compacted snow is stronger. In the model the strength is $σ = σ_{m} {(ρ / ρ_{m})}^{4}$ : snow can withstand pressure until it reaches the density of ice. The foot stops at the depth where $σ (z) = p$ — on the graph, the point where the curve crosses the pressure line. Simplifications: snow is compressed only downwards (it is not pushed sideways), the snow does not spring back when the foot is lifted, and the parameters are approximate.

Skis and snowshoes

A ski increases the contact area several times compared with a boot (about 5 times for a 1.6 m ski), and the pressure falls by the same factor. Fresh snow is compressed under a ski too, but at such a small pressure it is compressed much less and holds — a ski sinks far less than a boot. This is why the tracks of vehicles that work on marshy ground are wide, and heavy lorries have many wheels: increase the area to reduce the pressure.

Units of pressure

The SI unit is the pascal: $1 Pa = 1 {N/m}^{2}$ . This is a very small pressure — a sheet of paper lying on a table gives about 1 Pa, so the kPa and MPa are widely used. In engineering and everyday life non-SI units are also met: the bar ( $1 bar = 10^{5}$ Pa; the gauge pressure in a car tyre is usually about 2–2.5 bar), the standard atmosphere ( $1 atm = 101 325$ Pa), the millimetre of mercury ( $1 mmHg \approx 133.3$ Pa; barometers and blood-pressure monitors use this unit, $760 mmHg = 1$ atm), and the technical atmosphere ( $1 {kgf/cm}^{2} = 98 066.5$ Pa ≈ 0.98 bar). The “Pressure units” section of the panel shows the pressure under the foot in these units: for a boot only about 0.1 bar, but for the tip of a heel tens of bar.

The heel and the elephant

Let us calculate. Suppose a girl of 60 kg has a heel tip of 1 cm²; during a step the whole weight is on one heel: $p = 588.6 N / 0.0001 m^{2} \approx 5 886$ kPa. For the elephant we take approximate values: mass 4000 kg, each of its four feet a circle 40 cm in diameter: $S = 4 π \cdot {0.2}^{2} \approx 0.50$ m², $p = 39 240 N / 0.50 m^{2} \approx 78$ kPa. The pressure from the heel is about 75 times greater than from the elephant — which is why a thin heel leaves marks on parquet and linoleum.

Did you know?

The density of freshly fallen snow is usually about 50–200 kg/m³ — so roughly 80–95 % of its volume is air. That is why snow is a good thermal insulator.
When helping a person who has fallen through thin ice, rescuers move lying down or crawling on the ice: the contact area of a person lying down is about 10 times greater than that of the soles of the feet, so the pressure on the ice is that many times smaller.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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