Answer :
Final Answer:
The effective length of the G string on the nearby violin is 26.6 cm when a beat frequency of 2.00 Hz is heard between the two strings.
Explanation:
The beat frequency can be calculated using the equation f_beat = |f_1 - f_2|. The fundamental frequency of the G string on both violins is 196 Hz, so when a beat frequency of 2.00 Hz is heard, the frequency of the string on the nearby violin must be 198 Hz.
To find the effective length of the string on the nearby violin, we can use the formula f = (1/2L)*sqrt(T/μ), where L is the length of the string, T is the tension, and μ is the mass per unit length. We can rearrange this formula to solve for L, giving us L = (1/2)*(T/μ)*(1/f)^2.
Plugging in the values given, we find that the effective length of the string on the nearby violin is 26.6 cm. This is because shortening the string by sliding the finger down effectively increases the tension in the string, which causes the frequency to increase.
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Final answer:
The effective length of the G string is 9.8 cm.
Explanation:
To find the effective length of the string when the beat frequency is 2.00 Hz, we need to use the formula for the beat frequency: |f1 - f2|. First, we convert the beat frequency to the difference in frequencies, so we get |196 Hz - f2| = 2.00 Hz. Solving for f2, we find f2 = 196 Hz - 2.00 Hz = 194 Hz. Now we can use the formula for the fundamental frequency of a stretched string: f = (1/2L) * sqrt(T/μ), where L is the length of the string and μ is the linear mass density. Rearranging the formula to solve for L, we get L = (1/2) * (T/μ) * (1/f)^2. Plugging in the values, we have L = (1/2) * (0.500 g / (30.0 cm * 0.01 m/cm)) * (1/194 Hz)^2 = 0.098 m = 9.8 cm. Therefore, the effective length of the string is 9.8 cm.
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