Introduction
This question involves calculating the wave velocity and tension in a stretched string vibrating in its fundamental mode. We are given frequency, length, and mass of the string, and are asked to calculate the wave speed and tension.
Given
- Mass (m) = 20 g = 0.020 kg
- Length (L) = 40 cm = 0.40 m
- Frequency (f) = 30 Hz
- Amplitude (A) = 4 cm (not used in calculation of wave speed or tension)
Step 1: Calculate Wave Velocity
In the fundamental mode, the length of the string is equal to half the wavelength:
L = λ / 2 → λ = 2L = 2 × 0.40 = 0.80 m
Wave speed v is given by:
v = f × λ = 30 × 0.80 = 24 m/s
Step 2: Calculate Linear Density (μ)
μ = m / L = 0.020 / 0.40 = 0.05 kg/m
Step 3: Use Wave Speed to Find Tension
Wave speed is also given by:
v = √(T / μ) → T = μ × v²
T = 0.05 × 24² = 0.05 × 576 = 28.8 N
Final Answers
- Wave Velocity (v): 24 m/s
- Tension (T): 28.8 N
Conclusion
By applying wave formulas for a vibrating string in the fundamental mode, we calculated the wave speed and the tension in the string. These parameters are essential for understanding string dynamics and wave mechanics in musical instruments and other systems involving vibrating mediums.