The frequency of vibration of a violin string is given by f = (1/2L) √(T/rho), where L is the string length, T is tension, and rho is the linear density of the string. Find the rate of change of the frequency with respect to:
(a) The length (when T and rho are constant)
(b) The tension (when L and rho are constant)
(c) The linear density (when L and T are constant) The pitch of a note is determined by the frequency f. (The higher the frequency, the higher the pitch.) Use the signs of the derivatives in (a) through (c) to determine what happens to the pitch of a note:
(d) When the effective length of a string is decreased by placing a finger on the string so a shorter portion of the string vibrates.
(e) When the tension is increased by turning a tuning peg.
(f) When the linear density is increased by switching to another string.