Problem Statement
We aim to calculate the estimated lifespan of a pair of ballerina’s shoes given:
- Average body weight
- Common shoe materials
- Typical use frequency
- Wear and tear from dance maneuvers (jumps, turns, etc.)
We will find mathematical relationships among these parameters and illustrate the findings with graphs.
Variables and Assumptions
- \( W \): Body weight of the ballerina (assume 55 kg)
- \( F \): Average force exerted on shoes per jump or maneuver
- \( \mu \): Friction coefficient between shoes and floor (0.6, typical for dance floors)
- \( n \): Average number of dance maneuvers per session (e.g., 100 jumps, turns, etc.)
- \( t \): Number of sessions per week (assume 5 sessions per week)
- \( D \): Degradation rate of shoe material per force impact
- \( M \): Total material durability of the shoe (measured as a function of maximum cumulative force it can withstand before failing)
Mathematical Modeling of Shoe Lifespan
1. **Force Exerted per Maneuver**
The force exerted on the shoes during each maneuver can be calculated as:
\[
F = W \cdot g \cdot \mu
\]
where:
- \( g = 9.81 \, \text{m/s}^2 \): gravitational acceleration
- \( W \): body weight in kg
2. **Total Force Impact per Week**
Assuming \( n \) maneuvers per session and \( t \) sessions per week, the cumulative force on shoes per week is:
\[
F_{\text{weekly}} = F \cdot n \cdot t
\]
3. **Material Degradation**
Let \( D \) represent the material degradation rate, i.e., the proportion of the shoe's maximum durability capacity (\( M \)) lost per unit of force. Thus, the weekly degradation becomes:
\[
\text{Degradation}_{\text{weekly}} = F_{\text{weekly}} \cdot D
\]
4. **Lifespan Calculation**
The total lifespan \( L \) in weeks of the shoes can be estimated by dividing the total durability \( M \) by the weekly degradation:
\[
L = \frac{M}{\text{Degradation}_{\text{weekly}}} = \frac{M}{F \cdot n \cdot t \cdot D}
\]
Example Calculation with Assumptions
Assume typical values:
- \( W = 55 \, \text{kg} \)
- \( \mu = 0.6 \)
- \( n = 100 \) maneuvers per session
- \( t = 5 \) sessions per week
- \( D = 1 \times 10^{-5} \): degradation per unit force
- \( M = 500 \): durability threshold (cumulative force capacity before failure)
With these values, we can calculate \( L \) (in weeks) and plot it.
Graphical Representation of Shoe Lifespan Under Varying Parameters
The graph shows the relationship between body weight and shoe lifespan. Heavier weights result in greater forces and faster material degradation, reducing shoe lifespan.