The frequency of vibrations of a vibrating violin string is...

The frequency of vibrations of a vibrating violin string is given by $$ f=\frac{1}{2 L} \sqrt{\frac{T}{\rho}} $$ where $L$ is the length of the string. $T$ is its tension, and $\rho$ is its linear density. ISce Chapter 11 in $\mathbf{D}$. E. Hall, Musical Acousrics, 3 rd ed. (Pacific Grove, $\mathbf{C A}, 2002$ ). $.$ (a) Find the rate of change of the frequency with respect to (i) the length (when $T$ and $\rho$ are constant). (ii) the tension (when $L$ and $\rho$ are constant), and (iii) the linear density (when $L$ and $T$ are constant). (b) The pitch of a note (how high or low the note sounds) is determined by the frequency $f$. (The higher the frequency. the higher the pitch.) Use the signs of the derivatives in part (a) to determine what happens to the pitch of a note (i) when the effective length of a string is decreased by placing a finger on the string so a shorter portion of the string vibrates. (ii) when the tension is increased by turning a tuning $p e g_{1}$ (iii) when the linear density is increased by switching to another string.

The frequency of vibrations of a vibrating violin string is given by
$$
f=\frac{1}{2 L} \sqrt{\frac{T}{\rho}}
$$
where $L$ is the length of the string. $T$ is its tension, and $\rho$ is its linear density. ISce Chapter 11 in $\mathbf{D}$. E. Hall, Musical Acousrics, 3 rd ed. (Pacific Grove, $\mathbf{C A}, 2002$ ). $.$
(a) Find the rate of change of the frequency with respect to
(i) the length (when $T$ and $\rho$ are constant).
(ii) the tension (when $L$ and $\rho$ are constant), and
(iii) the linear density (when $L$ and $T$ are constant).
(b) The pitch of a note (how high or low the note sounds) is determined by the frequency $f$. (The higher the frequency. the higher the pitch.) Use the signs of the derivatives in
part (a) to determine what happens to the pitch of a note
(i) when the effective length of a string is decreased by placing a finger on the string so a shorter portion of the string vibrates.
(ii) when the tension is increased by turning a tuning $p e g_{1}$
(iii) when the linear density is increased by switching to another string.

Key Concepts

Physical Interpretation of Derivative Signs

The sign of a derivative provides insights into how the dependent variable responds to changes in the independent variable. A positive derivative indicates an increase, whereas a negative derivative implies a decrease. This interpretation is key to relating mathematical results back to physical phenomena, such as changes in pitch when string parameters are adjusted.

Sensitivity Analysis in Physical Models

This concept relates to examining how small changes in system parameters affect the outcome, such as the vibrational frequency of a string. Understanding this sensitivity helps in predicting how modifications like altering effective length, varying tension, or changing the string’s density will impact the system's properties.

Partial Differentiation

This concept involves taking the derivative of a function with multiple variables with respect to one variable while keeping the other variables constant. It is essential in understanding how one parameter, such as length, tension, or density, individually influences the system’s behavior.

Chain Rule

The chain rule is a fundamental technique in calculus used for differentiating composite functions, where one function is nested within another. In the context of the given frequency formula, it is applied to differentiate the square root segment with respect to its inner function.

Transcript

00:01 Hello, this question asks various things about the equation for frequency of notes on a string, which is 1 over 2l times the square root of t over rho.

00:11 So the first part asks what happens to the frequency when we change various factors in the equation.

00:20 So the first is length.

00:22 We're going to take a partial derivative here, which means that we're going to treat everything that is not the variable we're interested in as a constant.

00:31 So that's what it would be, del f del l.

00:34 We take the partial with respect to t, treat everything that's not t as a constant.

00:40 So 1 over 2l is a constant.

00:42 And then the derivative of the square root will be 1 half times t over rho to the negative 1 half.

00:50 And then we have to do chain rule as well.

00:52 So that's going to be times 1 over rho.

00:54 Simplifying that down, we get 1 over 4l rho times 1 over the square root of t over rho.

01:04 And then part three asks for if we change rho.

01:09 So again, treat everything that's not rho as a constant.

01:12 So the 1 over 2l is a constant.

01:14 Again, we're going to get the square root t over rho to the negative 1 half.

01:20 And then the chain rule here is a little more complicated...

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