Final answer:
To find the frequency of the string's fundamental mode of vibration, use the formula f₁ = (1/2L) * √(T/μ). Plugging in the given values, the frequency is approximately 127.8 Hz.
Step-by-step explanation:
To find the frequency of the string's fundamental mode of vibration, we can use the formula:
f₁ = (1/2L) * √(T/μ)
Where f₁ is the frequency, L is the length of the string, T is the tension in the string, and μ is the linear mass density of the string.
Plugging in the given values, we have:
f₁ = (1/2 * 0.700m) * √(765N / (5.25g/0.700m) * 9.8m/s²)
Convert the mass from grams to kilograms:
f₁ = (1/2 * 0.700m) * √(765N / (0.00525kg/0.700m) * 9.8m/s²)
Simplify the equation:
f₁ = (1/2 * 0.700m) * √(981600N / 0.00735kg)
f₁ = 350N * √133333.33
Calculate the square root:
f₁ ≈ 350N * 365.16 ≈ 127.8 Hz
Therefore, the frequency f₁ of the string's fundamental mode of vibration is approximately 127.8 Hz.