A violin string is 33cm long. The thinnest string on the violin is tuned to vibrate at a frequency of 659Hz.

(a) What is the wave velocity in the string?

(b) If you place your finger on the string so that its length is shortened to 28cm, what is the frequency of the note that the string produces?

λ =2•L = v/f.

v = 2•L•f = 2•0.33•659 = 465 m/s.
λ1 =2•L1 = v/f.
f = v/ 2•L1 = 465/2•0.28 =776.7 Hz.

To find the wave velocity in the string, we can use the formula:

Wave velocity (v) = Frequency (f) × Wavelength (λ)

(a) We know the frequency (f) is 659Hz. To find the wavelength (λ), we need to know the length of the string (L). The length of the string is given as 33cm.

Since the string is fixed at both ends, the wavelength is twice the length of the string:

λ = 2 × L

Substituting the given value of L:

λ = 2 × 33cm = 66cm

Now we can calculate the wave velocity:

v = f × λ = 659Hz × 66cm

Let's convert the wavelength from cm to meters to match the SI unit of frequency (Hz):

λ = 66cm × (1m/100cm) = 0.66m

v = f × λ = 659Hz × 0.66m = 434.94 m/s

Therefore, the wave velocity in the string is approximately 434.94 m/s.

(b) When you place your finger on the string, the effective length of the string changes. The new length is given as 28cm.

Using the formula for wave velocity (v):

v = f × λ

We want to solve for the new frequency (f). We already know the new length (L) of the string, which is 28cm.

To find the new wavelength (λ), we need to account for the change in length: λ = 2 × L.

Substituting the given value of L:

λ = 2 × 28cm = 56cm

Again, let's convert the wavelength from cm to meters:

λ = 56cm × (1m/100cm) = 0.56m

Now, we have the wavelength (λ) and the new length. We can rearrange the formula for wave velocity to solve for the new frequency:

f = v / λ

Substituting the known values:

f = 434.94m/s / 0.56m = 776.84Hz

Therefore, when the length of the string is shortened to 28cm, the string produces a note with a frequency of approximately 776.84Hz.