Elena, a physics student, is measuring the resonance frequency of a wire. She finds that tension F increases the frequency f according to f = (1/2L)√(F/μ), where L = 0.5 m, μ = 0.002 kg/m. If F increases from 40 N to 90 N, by how much does f increase?
["How Elena Measures Wire Resonance Frequency: Calculating the Increase in Sound Frequency When Tension Rises", "In her physics studies, Elena immerses herself in hands-on experiments to explore wave phenomena—one of which involves measuring the resonance frequency of a vibrating wire. Using the formula for resonance frequency, ( f = \frac{1}{2L} \sqrt{\frac{F}{\mu}} ), Elena investigates how changes in tension affect the sound frequency produced by a wire. With a wire length ( L = 0.5 ) meters and mass per unit length ( \mu = 0.002 , \ ext{kg/m} ), she applies increasing tension ( F ) to observe frequency shifts. This article explains her experiment, the underlying physics, and computes the frequency increase when tension rises from 40 N to 90 N.", "### The Physics Behind Resonance Frequency", "For a vibrating string fixed at both ends, the fundamental resonance frequency depends on tension, length, and linear mass density. The formula:\n[\nf = \frac{1}{2L} \sqrt{\frac{F}{\mu}}\n]\nshows that frequency increases with the square root of tension, given constant length and mass per unit length. This relationship explains why increasing tension sharpens pitch—higher tension raises wave speed on the wire, leading to greater oscillations per second.", "### Given Values", "- Wire length: ( L = 0.5 , \ ext{m} )\n- Mass per unit length: ( \mu = 0.002 , \ ext{kg/m} )\n- Initial tension: ( F_1 = 40 , \ ext{N} )\n- Final tension: ( F_2 = 90 , \ ext{N} )", "Using the formula, calculate the frequencies at both tensions:", "#### Step 1: Compute frequency at 40 N\n[\nf_1 = \frac{1}{2 \ imes 0.5} \sqrt{\frac{40}{0.002}} = \frac{1}{1} \sqrt{20,000} = \sqrt{20,000} \approx 141.42 , \ ext{Hz}\n]", "#### Step 2: Compute frequency at 90 N\n[\nf_2 = \frac{1}{1} \sqrt{\frac{90}{0.002}} = \sqrt{45,000} \approx 212.13 , \ ext{Hz}\n]", "### Frequency Increase", "Difference in frequency:\n[\n\Delta f = f_2 - f_1 = 212.13 - 141.42 = 70.71 , \ ext{Hz}\n]", "Thus, the resonance frequency increases by approximately 70.7 Hz when tension rises from 40 N to 90 N.", "### Conclusion", "Elena’s experiment vividly demonstrates how tension controls the pitch of a vibrating wire through the resonance frequency equation. Her measurements confirm that increasing tension significantly boosts frequency—offering insight into wave mechanics and acoustics. This simple yet powerful relationship underscores the beauty of physics in everyday sound phenomena.", "For students exploring wave behavior, Elena’s method highlights the importance of mathematical modeling and experimental verification—key pillars of scientific learning.", "---", "Keywords: resonance frequency, wire resonance, physics experiment, tension and frequency, wave speed, F = 1/(2L) √(F/μ), Elena physics student, sound frequency, wire vibration, L = 0.5 m, μ = 0.002 kg/m, frequency increase,科学实验, 物理学习, 纵波, 频率测量"]
