Answer :
To determine the vibrating section's length, the wave velocity on the string is calculated using the square root of the tension divided by the linear mass density. Subsequently, this velocity is used together with the known frequency for a violin's A string to solve for length, yielding approximately 52.34 centimeters.
The question is seeking the length of the vibrating section of a violin string under a given tension and linear density. To find the length, we must first determine the speed of the wave on the string using the formula for the velocity of a wave on a string, which is v = \\sqrt{(T / \\mu)}, where T is the tension and \\mu is the linear mass density. Once we have the velocity, we can use the fundamental frequency formula for a string fixed at both ends:
f₁ = v / 2L, where L is the length of the vibrating section of the string. For standard tuning, the fundamental frequency for the A string on a violin is 440 Hz. Solving for L gives us the length of the vibrating section.
To calculate the velocity (v), we convert the linear density from grams per meter to kilograms per meter by dividing by 1000. The tension (T) is already in newtons, the SI unit for force, so no further conversion is needed:
v = \\sqrt{(140 N / 0.00066 kg/m)}
= \\sqrt{(212121.2121 m^2/s^2)} \\approx 460.5648 m/s
Now, to find the length (L) given that the string is producing a frequency (f) of 440 Hz:
L = v / 2f = 460.5648 m/s / (2 \\times 440 Hz) \\approx 0.5234 m
Therefore, the vibrating section of the string is approximately 0.5234 meters or 52.34 centimeters long.
