A guitar string, destined to be an upper E string, has length 100 cm and mass 0.400 g.

a. What is its mass per unit length?
u = m/L = 0.004 g/cm

b. What length of this string should be used on a guitar if the fundamental frequency is to be 330 Hz and its tension is to be 100 N?

I'm not sure which equations to use for part B.

Nick,

I'm working on this one myself. You can deduce from V = sqrt (T/M) where M=mass/length that f*2L = sqrt (T/M) but I keep getting the wrong answer for L. Unit conversion maybe?

In order to find the length of the string, you can use the formula for the fundamental frequency of a vibrating string:

f = (1/2L) * sqrt(T/μ)

Where:
f is the fundamental frequency (in Hz),
L is the length of string (in meters),
T is the tension in the string (in Newtons), and
μ is the mass per unit length of the string (in kg/m).

To begin, you need to convert the given values into SI units:
For the mass per unit length, μ:
0.400 g = 0.400 * 10^(-3) kg
100 cm = 100 * 10^(-2) m

Now, you can substitute the given values into the formula and solve for L:

330 Hz = (1/2L) * sqrt(100 N / (0.400 * 10^(-3) kg/m))

First, let's rearrange the equation to isolate L:

L = sqrt((T/μ) / (4f^2))

Now, we can substitute the given values and solve for L:

L = sqrt((100 N / (0.400 * 10^(-3) kg/m)) / (4 * 330 Hz)^2)

Using your calculator, you can simplify and solve for L:

L ≈ sqrt(625 / 43560) ≈ 0.044679 m

Therefore, the length of the string that should be used on a guitar to achieve a fundamental frequency of 330 Hz and a tension of 100 N is approximately 0.044679 meters (or about 4.4679 cm).