A telegraph wire made of copper is stretched tightly between two telephone poles 50 m apart. The tension in the wire is 500 N, and the mass per unit length is 2.0 × 10−2 kg/m. What is the frequency of the fundamental mode and what is the first overtone?
The correct answer and explanation is:
To calculate the frequency of the fundamental mode and the first overtone of a stretched wire, we need to use the formula for the fundamental frequency of a vibrating string under tension:
Fundamental Frequency:
The formula for the frequency of the fundamental mode is: f1=12LTμf_1 = \frac{1}{2L} \sqrt{\frac{T}{\mu}}
where:
- f1f_1 is the fundamental frequency.
- LL is the length of the wire (50 m in this case).
- TT is the tension in the wire (500 N).
- μ\mu is the mass per unit length of the wire (2.0×10−22.0 \times 10^{-2} kg/m).
Step 1: Calculate the Fundamental Frequency
First, we need to calculate the fundamental frequency using the given data: f1=12LTμf_1 = \frac{1}{2L} \sqrt{\frac{T}{\mu}}
Substituting the values: f1=12×505002.0×10−2f_1 = \frac{1}{2 \times 50} \sqrt{\frac{500}{2.0 \times 10^{-2}}} f1=110025,000f_1 = \frac{1}{100} \sqrt{25,000} f1=1100×158.11f_1 = \frac{1}{100} \times 158.11 f1≈1.58 Hzf_1 \approx 1.58 \, \text{Hz}
Step 2: Calculate the First Overtone
The frequency of the first overtone (also known as the second harmonic) is given by: f2=2f1f_2 = 2f_1
Thus: f2=2×1.58f_2 = 2 \times 1.58 f2≈3.16 Hzf_2 \approx 3.16 \, \text{Hz}
Final Answer:
- The frequency of the fundamental mode is approximately 1.58 Hz.
- The frequency of the first overtone is approximately 3.16 Hz.
Explanation:
The fundamental frequency corresponds to the lowest frequency at which the wire vibrates, and this is determined by the tension, length, and mass per unit length of the wire. The first overtone (second harmonic) is simply double the fundamental frequency because the wire vibrates at twice the frequency of the fundamental mode in this overtone. The tension in the wire influences the speed of the wave propagation, which in turn determines the frequencies of vibration for different modes.