Question
How do I set up a standing wave with a frequency of 455 Hz and 5 antinodes on a 2.49-m-long string with a mass of 7.17 g?
Answers
GPT-4o mini
To set up a standing wave on a string, you'll need to understand the relationship between frequency, wave speed, and wavelength. For a string of length \( L \) with \( n \) antinodes, the relationship can be described mathematically as follows:
1. **Standing Wave Conditions**:
- The length \( L \) of the string corresponds to \( n \) antinodes; hence, it fits \( n/2 \) wavelengths into the length of the string. Therefore, the relationship is given by:
\[
L = \frac{n}{2} \lambda
\]
- Rearranging gives:
\[
\lambda = \frac{2L}{n}
\]
2. **Calculating Wavelength**:
- Given \( L = 2.49 \) m and \( n = 5 \):
\[
\lambda = \frac{2 \times 2.49 \, \text{m}}{5} = \frac{4.98 \, \text{m}}{5} = 0.996 \, \text{m}
\]
3. **Wave Speed**:
- The wave speed \( v \) is related to frequency \( f \) and wavelength \( \lambda \) by the equation:
\[
v = f \lambda
\]
- Substituting \( f = 455 \) Hz and \( \lambda = 0.996 \) m:
\[
v = 455 \, \text{Hz} \times 0.996 \, \text{m} \approx 453.78 \, \text{m/s}
\]
4. **Tension in the String**:
- The wave speed \( v \) can also be expressed in terms of the tension \( T \) in the string and the mass per unit length \( \mu \):
\[
v = \sqrt{\frac{T}{\mu}}
\]
- The mass per unit length \( \mu \) is calculated as the total mass divided by the length of the string:
\[
\mu = \frac{m}{L} = \frac{7.17 \, \text{g}}{2.49 \, \text{m}} = \frac{0.00717 \, \text{kg}}{2.49 \, \text{m}} \approx 0.00288 \, \text{kg/m}
\]
5. **Finding Tension**:
- Rearranging and squaring both sides gives:
\[
T = \mu v^2
\]
- Plugging in the values:
\[
T = 0.00288 \, \text{kg/m} \times (453.78 \, \text{m/s})^2 \approx 0.00288 \times 205,925.5 \approx 593.0 \, \text{N}
\]
### Summary
To set up your standing wave with a frequency of 455 Hz and 5 antinodes on a 2.49 m-long string:
- You need to ensure the tension in the string is approximately **593 N**.
- Use a wave generator that can produce a frequency of **455 Hz**.
- Make sure the string is fixed at both ends and properly tensioned. Depending on your setup, use a tuning mechanism to adjust the tension as needed.
With the frequency generator set to 455 Hz and the string tension adjusted to approximately 593 N, you should be able to establish the desired standing wave.
1. **Standing Wave Conditions**:
- The length \( L \) of the string corresponds to \( n \) antinodes; hence, it fits \( n/2 \) wavelengths into the length of the string. Therefore, the relationship is given by:
\[
L = \frac{n}{2} \lambda
\]
- Rearranging gives:
\[
\lambda = \frac{2L}{n}
\]
2. **Calculating Wavelength**:
- Given \( L = 2.49 \) m and \( n = 5 \):
\[
\lambda = \frac{2 \times 2.49 \, \text{m}}{5} = \frac{4.98 \, \text{m}}{5} = 0.996 \, \text{m}
\]
3. **Wave Speed**:
- The wave speed \( v \) is related to frequency \( f \) and wavelength \( \lambda \) by the equation:
\[
v = f \lambda
\]
- Substituting \( f = 455 \) Hz and \( \lambda = 0.996 \) m:
\[
v = 455 \, \text{Hz} \times 0.996 \, \text{m} \approx 453.78 \, \text{m/s}
\]
4. **Tension in the String**:
- The wave speed \( v \) can also be expressed in terms of the tension \( T \) in the string and the mass per unit length \( \mu \):
\[
v = \sqrt{\frac{T}{\mu}}
\]
- The mass per unit length \( \mu \) is calculated as the total mass divided by the length of the string:
\[
\mu = \frac{m}{L} = \frac{7.17 \, \text{g}}{2.49 \, \text{m}} = \frac{0.00717 \, \text{kg}}{2.49 \, \text{m}} \approx 0.00288 \, \text{kg/m}
\]
5. **Finding Tension**:
- Rearranging and squaring both sides gives:
\[
T = \mu v^2
\]
- Plugging in the values:
\[
T = 0.00288 \, \text{kg/m} \times (453.78 \, \text{m/s})^2 \approx 0.00288 \times 205,925.5 \approx 593.0 \, \text{N}
\]
### Summary
To set up your standing wave with a frequency of 455 Hz and 5 antinodes on a 2.49 m-long string:
- You need to ensure the tension in the string is approximately **593 N**.
- Use a wave generator that can produce a frequency of **455 Hz**.
- Make sure the string is fixed at both ends and properly tensioned. Depending on your setup, use a tuning mechanism to adjust the tension as needed.
With the frequency generator set to 455 Hz and the string tension adjusted to approximately 593 N, you should be able to establish the desired standing wave.