Final answer:
The violinist has slid her finger down by approximately 0.53 cm on the G string to create the 3.5 Hz beat frequency when compared with another G string of the same violin vibrating at 196 Hz.
Step-by-step explanation:
To find out how far down the string the violinist slid her finger to create a beat frequency of 3.5 Hz, we first need to determine the frequency of the second vibrating string.
Knowing that one violin string is already vibrating at a fundamental frequency of 196 Hz, we can calculate the frequency of the second string by adding or subtracting the beat frequency, as beat frequency is the absolute difference between the two frequencies.
Thus, the frequency of the second string could be either f2 = 196 Hz + 3.5 Hz or f2 = 196 Hz - 3.5 Hz. Let's choose the higher frequency for this instance, so we have f2 = 199.5 Hz.
Next, we use the formula for the fundamental frequency of a string, which is f1 = v/(2L), where f1 is the fundamental frequency, v is the wave speed on the string, and L is the length of the string.
Since the string lengths are initially identical and the wave speeds are the same on both strings, we can express the second length, L2, as L0 - x, where L0 is the initial length and x is the length the violinist's finger has covered.
Applying the fact that the frequencies are inversely proportional to the lengths (assuming string tension and mass per unit length remain constant), we can set up the following equation: f1/f2 = (L0 - x) / L0.
Now, we calculate the length the string has been shortened: 196 Hz / 199.5 Hz = (30.0 cm - x) / 30.0 cm. Solving for x yields x = 30.0 cm (1 - 196/199.5), which results in x = 0.53 cm. Therefore, the violinist has slid her finger down the string by 0.53 cm to create the 3.5 Hz beat frequency.