### Slippin' and Slidin'

March 29, 2021

Ice skating has given us some easily understood examples of a few physical principles. The principal one, of course, is the concept of the conservation of angular momentum. As often seen in performance, when a rotating figure skater with arms extended pulls her arms inward, her rotational speed increases. Her moment of inertia I is decreased when her arms are placed inwards, so the rotational speed ω is increased in order to keep a constant angular momentum L; viz.,
L = Iω

Figure skating example of the conservation of angular momentum. The closest example to the relevant moment of inertia for a skater who uses the movement of her arms to change her inertia is that of a cylinder rotating about its center, as shown on the left. The moment of inertia L of an idealized case for a cylinder of mass m and length is L = (1/12)mℓ2. Since the length appears as a square, the change of inertia with length is large. The factor of (1/12) appears strange at first, but it's just a consequence of an integration of point masses along the cylinder. Shown on the right is American figure skater, Sasha Cohen (b. 1984), the 2006 Olympic silver medalist, preparing for a spin. (Left, a modified Wikimedia Commons image by Krishnavedala. Right image, Sasha Cohen at the qualification program at the World Figure Skating Championships, 2004, in Dortmund, by Uwe Langer. Click for larger image.)

Although the icy surface of a skating rink looks like glass, you can't skate on glass because of friction. The coefficient of friction depends on what materials are rubbing against each other. The coefficient of dynamic friction between the material "couple" of iron or steel on glass is a little more than 0.5,[1] but that for steel on ice is less than a tenth of this value. What makes ice so slippery?

While it's apparent that a lubricating layer of liquid water is responsible for the low friction, the method by which this layer forms has been debated for more than a century. In one theory, the initial friction of the skate blade heats the ice to melt it.[2-3] In another, it's the pressure on the blade that causes the melting. This is an expected consequence of the fact that ice is less dense than water. A transformation from ice to liquid water reduces the volume and increases the temperature according to the Clausius–Clapeyron relation for ice,
(∂P/∂T) = -13.5 mPa/K = 1958 psi/°C
in which P is the pressure, and T is the temperature. It's good that skate blades are sharp, since an ice layer 5 °C below freezing needs 10,000 psi (pounds per square inch) for melting. For most other materials, the volume increases upon melting, so increased pressure will increase the melting point. A good review of the physics of ice skating can be found at ref.4.[4]

A team of physicists from the University of Amsterdam (Amsterdam, Netherlands) has just published an open access paper in Physical Review X in which they report on their ice experiments.[5-6] They show that the hardness of ice decreases with increasing temperature; and, when the ice hardness is below the contact pressure of a pressed sphere, how the sphere will undergo increased friction by ploughing the surface. This ploughing action causes a temperature increase that further decreases the hardness.[5] They find that the ease of sliding on ice has a strong dependence on temperature, contact pressure, and speed.[5-6]

Density of ice as a function of temperature.

Although liquid water expands as it freezes into ice, from that point ice behaves as most solids by becoming denser as temperature decreases.

(Data from the Engineering Toolbox, graphed using Gnumeric. Click for larger image.)

Previous research put an emphasis on the layer of water layer interposed between the sliding object and the surface of the ice, but this layer alone doesn't explain why friction is higher near the melting point of ice.[6] The Netherlands' researchers did experiments in a temperature range from -120 to -1.5 °C on a very smooth ice surface prepared by repeatedly adding a fresh water film, effectively doing a Zamboni treatment.[5-6] They measured the friction using a rheometer with different slider shapes - a large sphere, a small sphere, and a shape similar to a skate blade.[5-6] They also measured the hardness in a manner similar to the Rockwell method.[5-6]

The ice friction follows an Arrhenius law behavior at temperatures far below its melting point.[5] Although its hardness decreases near the melting point, ice is still hard enough for sliding.[5-6] This performance is different from that of most other materials, which become soft near their melting point and don't allow easy sliding.[6] Near the higher friction regime of ice near its melting point, blades penetrate this softer surface and start to plough through it.[5-6] The different shapes start to plough at different temperatures, with the small sphere ploughing at about -20°C, and the skate blade shape ploughing at -8°C because of its lower contact pressure.[6] The experiments also showed that the ice hardness increases with slider speed; so, a fast skate should plough less and thus slide better.[6]

Slipping and ploughing regimes for ice.

As the temperature is increased towards melting, a point will be reached where the ice hardness is below the contact pressure of the slider. At that point, the slider will plough though the ice surface, causing more friction.

(Created using Inkscape.)

"Slippin' and Slidin'" is a 1955 song co-written and performed by Little Richard (Richard Penniman (1932-2020)) and released in other versions in subsequent years.[7]

### References:

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