Suppose you have a string that is being made to vibrate at a specific frequency, f. The speed of waves on the string is v. The length of the string that is vibrating, L, is adjustable. When the length of the string has been adjusted so that the fundamental standing wave pattern forms, write the expression for L in terms of f and v. (The fundamental standing wave pattern corresponds to nodes at each end of the string and an antinode in the middle.)

L = ?

Is L = v/f the correct answer?

wavelength = twice string length to get node at each end and bulge in middle.

L = (1/2) wavelength

wavelength = speed * period = speed/frequency

so

L = (1/2) v/f

No, L = v/f is not the correct answer. To find the correct expression for L in terms of f and v, we can use the formula for the speed of a wave on a string:

v = f * λ

where v is the speed of the waves on the string, f is the frequency of the wave, and λ is the wavelength.

In the case of the fundamental standing wave pattern, the length of the vibrating string, L, is equal to half of the wavelength (since there is a node at each end of the string and an antinode in the middle). Mathematically, we can represent this as:

L = λ/2

To find the value of L in terms of f and v, we need to eliminate λ from the equation. We can do this by rearranging the equation for the speed of waves:

λ = v / f

Now, substitute the value of λ in the expression for L:

L = (v / f) / 2

Simplifying further, we get:

L = v / (2f)

Therefore, the correct expression for L in terms of f and v is L = v / (2f).