Required information When a person ice-skates, the ice surface actually melts beneath the blades so that he or she skates on a thin film of water between the blade and the ice. Find an expression for the total friction force Fon the bottom of the blade as a function of skater velocity V, blade length L, water film thickness h, water viscosity, and blade width W.
Required information When a person ice-skates, the ice surface actually melts beneath the blades so that he or she skates on a thin film of water between the blade and the ice. Find an expression for the total friction force Fon the bottom of the blade as a function of skater velocity V, blade length L, water film thickness h, water viscosity, and blade width W.
Chapter2: Loads On Structures
Section: Chapter Questions
Problem 1P
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![## Understanding Friction in Ice Skating
**Required Information:**
When a person ice-skates, the ice surface actually melts beneath the blades, creating a thin film of water between the blade and the ice. This phenomenon reduces friction and allows the skater to glide smoothly across the ice.
### Problem Statement:
**Objective:**
Find an expression for the total friction force \( F \) on the bottom of the blade as a function of skater velocity \( V \), blade length \( L \), water film thickness \( h \), water viscosity \( \mu \), and blade width \( W \).
This scenario can be modeled using principles of fluid dynamics and friction forces. The friction force in this context arises due to the viscous shear stress in the thin film of water.
### Key Variables:
- **\( V \)**: Velocity of the skater.
- **\( L \)**: Length of the blade.
- **\( h \)**: Thickness of the water film.
- **\( \mu \)**: Viscosity of the water.
- **\( W \)**: Width of the blade.
### Diagram Explanation:
(There is no provided graph or diagram in the image, so we’ll explain how this system could be visualized.)
Imagine a side view of an ice skate blade gliding on the ice surface. The blade is in contact with the ice through a very thin layer of water. This water layer is a result of the heat generated by the pressure and friction of the blade which causes the ice to melt temporarily.
### Expression for Friction Force:
The friction force \( F \) due to the viscous drag in the water film can be determined using the formula:
\[ F = \text{Shear stress} \times \text{Contact area} \]
The shear stress \( \tau \) in a fluid layer can be expressed as:
\[ \tau = \mu \left( \frac{dV}{dz} \right) \]
Where \( \frac{dV}{dz} \) is the velocity gradient perpendicular to the flow direction (i.e., across the water film thickness \( h \)).
In our case, \( \frac{dV}{dz} \approx \frac{V}{h} \). Hence, the shear stress \( \tau \) is:
\[ \tau = \mu \left( \frac{V}{h} \right) \]
The contact area \( A](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbab71d29-f3d9-43ce-83f4-a263357daa0d%2Fe03a5a03-63e3-4e99-8944-85a9e82371c0%2Fwevikaq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Understanding Friction in Ice Skating
**Required Information:**
When a person ice-skates, the ice surface actually melts beneath the blades, creating a thin film of water between the blade and the ice. This phenomenon reduces friction and allows the skater to glide smoothly across the ice.
### Problem Statement:
**Objective:**
Find an expression for the total friction force \( F \) on the bottom of the blade as a function of skater velocity \( V \), blade length \( L \), water film thickness \( h \), water viscosity \( \mu \), and blade width \( W \).
This scenario can be modeled using principles of fluid dynamics and friction forces. The friction force in this context arises due to the viscous shear stress in the thin film of water.
### Key Variables:
- **\( V \)**: Velocity of the skater.
- **\( L \)**: Length of the blade.
- **\( h \)**: Thickness of the water film.
- **\( \mu \)**: Viscosity of the water.
- **\( W \)**: Width of the blade.
### Diagram Explanation:
(There is no provided graph or diagram in the image, so we’ll explain how this system could be visualized.)
Imagine a side view of an ice skate blade gliding on the ice surface. The blade is in contact with the ice through a very thin layer of water. This water layer is a result of the heat generated by the pressure and friction of the blade which causes the ice to melt temporarily.
### Expression for Friction Force:
The friction force \( F \) due to the viscous drag in the water film can be determined using the formula:
\[ F = \text{Shear stress} \times \text{Contact area} \]
The shear stress \( \tau \) in a fluid layer can be expressed as:
\[ \tau = \mu \left( \frac{dV}{dz} \right) \]
Where \( \frac{dV}{dz} \) is the velocity gradient perpendicular to the flow direction (i.e., across the water film thickness \( h \)).
In our case, \( \frac{dV}{dz} \approx \frac{V}{h} \). Hence, the shear stress \( \tau \) is:
\[ \tau = \mu \left( \frac{V}{h} \right) \]
The contact area \( A
Transcribed Image Text:**Question:**
Which of the following equations correctly represents the relationship among the given variables?
**Options:**
1. \( \circ \quad F = \frac{\mu VLW}{h} \)
2. \( \circ \quad F = \frac{VLW}{\mu^2 h} \)
3. \( \circ \quad F = \mu V^2 LWh \)
4. \( \circ \quad F = \mu VLWh \)
Please select the correct formula based on the context provided.
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