Question

The frequency of vibrations of a vibrating violin string is given by

f= 1/2L √T/P
where L is the length of the string, T is its tension, and pis its linear density. Find the rate of change of the frequencywith respect to
a) the length (when T and P areconstant)
b) the tension (when L and P areconstant)
c) the linear density (when L and T areconstant)

Answer 1

To find the rate of change of the frequency with respect to different variables, differentiate the formula f = 1/2L √T/P. The rates of change are df/dL = -1/2L^2 √T/P, df/dT = 1/2L √1/T∙P, and df/dP = -1/2L √T/P^2.

Step-by-step explanation:

To find the rate of change of the frequency with respect to:

a) the length (when T and P are constant), differentiate the formula f = 1/2L √T/P with respect to L. The rate of change is given by df/dL = -1/2L^2 √T/P.

b) the tension (when L and P are constant), differentiate the formula f = 1/2L √T/P with respect to T. The rate of change is given by df/dT = 1/2L √1/T∙P.

c) the linear density (when L and T are constant), differentiate the formula f= 1/2L √T/P with respect to P. The rate of change is given by df/dP = -1/2L √T/P^2.