A violin string vibrates at 300 Hz when unfingered. At what frequency will it vibrate if it is fingered one-third of the way down from the end? (That is, only two-thirds of the string vibrates as a standing wave.)
300
To determine the frequency at which the violin string will vibrate when fingered one-third of the way down, we need to understand the relationship between the length of a vibrating string and its frequency.
The frequency of vibration (f) for a string is inversely proportional to the length (L) of the string, according to the formula:
f ∝ 1/L
Now, let's denote the original length of the string as L0 and the length of the fingered portion as L1 (which is two-thirds of the original length). We want to find the frequency (f1) when L1 is used.
Given that the frequency when unfingered (f0) is 300 Hz, we can set up the equation:
f0 = 1 / L0
To find f1, we need to determine the length L1 based on the original length L0. Since the length of the fingered portion is one-third of the total length, we have:
L1 = (1/3) * L0
Now, let's substitute this value into our equation:
f1 = 1 / L1
Since L1 = (1/3) * L0, we get:
f1 = 1 / [(1/3) * L0]
Simplifying this equation, we have:
f1 = 3 / L0
Therefore, the frequency when the string is fingered one-third of the way down is three times the frequency when unfingered. To find the exact value for f1, multiply f0 (300 Hz) by 3:
f1 = 3 * f0
f1 = 3 * 300 Hz
f1 = 900 Hz
So, when the string is fingered one-third of the way down, it will vibrate at a frequency of 900 Hz.