Calculation of Coefficient of Friction between Ice and Figure Skating Blades

The frictional force between the ice and the blades of a figure skater is determined by several factors, including the weight of the skater, the properties of the ice, and environmental conditions like humidity and pressure. We will derive the mathematical relationships that influence the coefficient of friction ($\mu$) in this scenario.

1. Problem Statement

A figure skater moves on the ice while wearing skates with metal blades. The primary forces at play are the skater's weight and the frictional force exerted between the ice and the blades. The coefficient of friction ($\mu$) quantifies the amount of resistance to motion between two surfaces. We aim to find the relationship between $\mu$, the skater's weight, the properties of the ice, and environmental factors like pressure and humidity.

2. Standard Parameters

• Weight of the skater: $W$, assumed to be an average built athlete's weight (e.g., 70 kg).
• Humidity: $H$, affecting the melting point of ice and thus the layer of water between the blades and the ice.
• Pressure: $P$, influencing how the pressure of the blades compresses the ice and alters friction.
• Temperature: $T$, which also affects the ice's physical state and friction coefficient.
• Area of the blade in contact with the ice: $A$.

3. Basic Frictional Force

The frictional force $F_{\text{friction}}$ is given by:

$$F_{\text{friction}}=\mu N$$

where $\mu$ is the coefficient of friction and $N$ is the normal force, which, in the case of a flat surface, equals the weight of the skater $N=W$.

Therefore, the frictional force can be expressed as:

$$F_{\text{friction}}=\mu W$$

4. Relationship of Coefficient of Friction with Environmental Factors

The coefficient of friction $\mu$ is not constant and can vary depending on several factors:

4.1. Humidity

Humidity $H$ affects the layer of water that forms between the ice and the blade. Higher humidity leads to more water, reducing the friction by creating a lubricating layer. The relationship between $\mu$ and $H$ can be modeled as:

$$\mu (H)={\mu }_0\cdot e^{-kH}$$

where ${\mu }_0$ is the coefficient of friction at zero humidity, and $k$ is a constant that depends on the ice properties and blade material.

4.2. Pressure

The pressure $P$ applied by the skater's blades also plays a significant role in the friction between the blade and the ice. The pressure alters the ice's state, affecting the friction. This can be described by the following relationship:

$$\mu (P)={\mu }_0\cdot (1+cP)$$

where $c$ is a constant that depends on the ice type and blade material, and $P$ is the pressure exerted by the skater's weight over the blade area.

4.3. Temperature

Temperature $T$ affects the ice's properties and the lubrication between the blade and the ice. At lower temperatures, ice is harder and offers more resistance, resulting in a higher coefficient of friction. As the temperature increases, the ice becomes softer, and a thin layer of water forms, reducing friction. The relationship can be modeled as:

$$\mu (T)={\mu }_0\cdot e^{-dT}$$

where $d$ is a constant that represents how temperature changes the coefficient of friction, and $T$ is the temperature of the ice in Celsius.

5. Complete Equation for Coefficient of Friction

The total coefficient of friction can be derived by considering the effects of humidity, pressure, and temperature simultaneously. Combining all the factors together, we get the following expression:

$$\mu ={\mu }_0\cdot e^{-kH}\cdot (1+cP)\cdot e^{-dT}$$

Here:

• ${\mu }_0$ is the baseline coefficient of friction at zero humidity, pressure, and temperature.
• $H$ is the humidity.
• $P$ is the pressure exerted by the skater's weight.
• $T$ is the temperature of the ice.
• $k,c,d$ are constants that depend on the material properties of the ice and the blades.

6. Conclusion

The coefficient of friction between the blades of a figure skater and the ice is influenced by environmental factors such as humidity, pressure, and temperature, as well as the skater’s weight. By understanding and calculating these relationships, we can optimize the skater’s performance by reducing the friction as much as possible under the right conditions. The derived equation gives a comprehensive way to model this frictional interaction.