Answer :
To keep the violin string in tune with the organ pipe when the temperature increases from 0°C to 20°C, the tension in the violin string needs to increase by 0.04%.
To find the change in tension required to keep the violin string in tune with the organ pipe when the temperature increases from 0°C to 20°C, we'll use the formula relating tension, frequency, and length of the string.
Given:
- The initial temperature [tex]\( T_{\text{initial}} = 0^\circ C \)[/tex]
- The final temperature [tex]\( T_{\text{final}} = 20^\circ C \)[/tex]
- Coefficient of thermal expansion [tex](\( \alpha \))[/tex] for the violin string material
Let's proceed with the calculations:
1. Calculate the fractional change in temperature:
[tex]\[ \Delta T = T_{\text{final}} - T_{\text{initial}} = 20^\circ C - 0^\circ C = 20^\circ C \][/tex]
2. Use the coefficient of thermal expansion:
Assuming the coefficient of thermal expansion [tex](\( \alpha \))[/tex] for the violin string material is known, we can use it to find the fractional change in tension.
[tex]\[ \frac{\Delta T}{T} = \alpha \times \Delta T \][/tex]
3. Calculate the new tension:
[tex]\[ T_{\text{new}} = T_{\text{initial}} + \Delta T \][/tex]
Substitute the known values into the formulas.
Now, let's proceed with an example calculation:
Suppose the coefficient of thermal expansion [tex](\( \alpha \))[/tex] for the violin string material is [tex]\( 2 \times 10^{-5} \, \text{K}^{-1} \).[/tex]
[tex]\[ \frac{\Delta T}{T} = (2 \times 10^{-5}) \times 20 \][/tex]
[tex]\[ \frac{\Delta T}{T} = 4 \times 10^{-4} \][/tex]
[tex]\[ \frac{\Delta T}{T} = 0.0004 \][/tex]
So, the tension in the violin string needs to increase by [tex]\( 0.04\% \)[/tex] to stay in tune with the organ pipe.
