Suppose that the strings on a violin are stretched with the...

Suppose that the strings on a violin are stretched with the same tension and each has the same length between its two fixed ends. The musical notes and corresponding fundamental frequencies of two of these strings are $\mathrm{G}(196.0 \mathrm{~Hz})$ and $\mathrm{E}(659.3 \mathrm{~Hz})$. The linear density of the E string is $3.47 \times 10^{-4} \mathrm{~kg} / \mathrm{m} .$ What is the linear densitv of the G strine?

Key Concepts

Vibrating String Frequency Formula

This concept involves the formula for the fundamental frequency of a vibrating string fixed at both ends: f = (1/(2L))?(T/?), where f is the frequency, L is the length of the string, T is the tension, and ? is the linear density. It explains how the frequency is determined by the physical properties of the string.

Dependence on Linear Density

Under constant tension and fixed string length, the formula shows that the frequency is inversely proportional to the square root of the linear density (?). This means that if the linear density increases, the frequency decreases, and vice versa, which is central to understanding how different strings produce different pitches.

Proportional Reasoning in Physical Systems

This concept relates to using ratios and proportional relationships in physics problems. By comparing two strings with the same tension and length, one can derive relationships between their frequencies and linear densities, allowing for the determination of an unknown quantity when the corresponding ratio is known.

Transcript

0:00 Hi there.

00:01 So for this problem, we are given the fundamental frequencies for two streams of a violin.

00:07 So we are given the fundamental frequency of the e string.

00:12 It's 659 .3 hertz.

00:17 And the fundamental frequency for the g string is 196 hertz.

00:22 So we also given the linear density of the e string, and that means the mass over the length of that e of string, and that is 3 .47 times 10 to the minus 4 kilograms per meter.

00:37 So with this information, what we need to find is the linear density of the g string.

00:51 Now, we're going to start by knowing that the fundamental frequency of a string, in this case, when n equals to 1, for a string that is fits it at both n, it is the following.

01:08 The fundamental frequency is equal to the speed of the weight over two times the length.

01:19 Now, we also note that the force that is being applied in a string is related through the speed, and that is the speed is equal to the square root of the force applied over the linear density.

01:42 So what we can do in here is to solve in here for the velocity or the speed and just equal these two expressions.

01:54 So to do that, we obtain that the speed is equals to two times the length of the string times the fundamental frequency.

02:03 So we solve that in here.

02:19 Now we need to find a weight so that we can obtain that because the force that we are applying to both strings are the same and the length of this of these two streams are the same.

02:35 So what we need to do is to rearrange this in a convenient in 4.

02:40 So as you can see, what we can do is to elevate this to the squared in order to eliminate the square root of the right side.

02:54 So we will obtain that this is this to, and this is equal to to times the length, the frequency.

03:07 So what we can do in here, as you can see, and we can rearrange this in the following form.

03:20 In the following form, we can obtain that the force that is being applied over the length squared is equal to, well, in here, and remember that i elevate this to the square...

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