# Thread: Rate of change of frequency of vibrations of a violin string?

1. ## Rate of change of frequency of vibrations of a violin string?

We're learning related rates/rates of change currently in my AP calc class. Just have a few questions, wondering if I'm understanding these problems correctly. The " " is my questions, etc.

f = 1/2L * sq. root ( T / P )

The frequency of vibrations of a violin string is given by the above eq. where L is the length of the string, T is its tension, and P is linear density.

#1. Find rate of change of f with respect to: a) the length (when T & P are constant), b) the tension (when T & P are constant), c) the linear density (when T & P are density)

" Do I take the first three derivatives of f to find these? First deriv being length, second being tension, third being linear density? "

#2 The pitch of a note (how high or low the note sounds) is determined by the frequency F. Use the signs of the derivatives in #1 to determine what happens to the pitch of a note:

" I don't understand this at all "

a) when the effective length of a string is decreased by placing a finger on the string so a shorter portion of the string vibrates

b) when the tension is increased by turning a tuning peg

c) when the linear density is increased by switching to another string

2. Originally Posted by arkhampatient
We're learning related rates/rates of change currently in my AP calc class. Just have a few questions, wondering if I'm understanding these problems correctly. The " " is my questions, etc.

f = 1/2L * sq. root ( T / P )

The frequency of vibrations of a violin string is given by the above eq. where L is the length of the string, T is its tension, and P is linear density.

#1. Find rate of change of f with respect to: a) the length (when T & P are constant), b) the tension (when T & P are constant), c) the linear density (when T & P are density)

" Do I take the first three derivatives of f to find these? First deriv being length, second being tension, third being linear density? "
No you take the derivative of f with respect to a) L, b) T, and c) P.

In each case you treat the other variables as if they were constants.

CB

3. Originally Posted by arkhampatient

#2 The pitch of a note (how high or low the note sounds) is determined by the frequency F. Use the signs of the derivatives in #1 to determine what happens to the pitch of a note:

" I don't understand this at all "

a) when the effective length of a string is decreased by placing a finger on the string so a shorter portion of the string vibrates

b) when the tension is increased by turning a tuning peg

c) when the linear density is increased by switching to another string
When you shorten the string the frequency becomes:

$\displaystyle f(L-\varepsilon,T,P) \approx f(L,T,P) -\varepsilon f_L(L,T,P)$

where $\displaystyle f_L(L,T,P)$ denotes the derivative of $\displaystyle f(L,T,P)$ with respect to $\displaystyle L$ evaluated at $\displaystyle (L,T,P)$.

Part a) asks: Is this an increase or decrease in frequency?

The others are similar except that the variables in question are increased rather than decreased.

CB