String Vibrations: Determine Wavelength & Fundamental Frequency

[Husker70]

Homework Statement

A standing wave is established in a 120-cm-long string fixed at both
ends. The string vibrates in four segments when driven at 120Hz.
(a) Determine the wavelength
(b) What is the fundamental frequency of the string?

Homework Equations

(a) Lambda = 2L/n

The Attempt at a Solution

(a) Lambda = 2(120cm) / 4 = 60cm

Is this right? doesn't seem to be
Kevin

 

[alphysicist]

Hi Husker70,

Homework Statement

A standing wave is established in a 120-cm-long string fixed at both
ends. The string vibrates in four segments when driven at 120Hz.
(a) Determine the wavelength
(b) What is the fundamental frequency of the string?

Homework Equations

(a) Lambda = 2L/n

The Attempt at a Solution

(a) Lambda = 2(120cm) / 4 = 60cm

Is this right? doesn't seem to be

That looks right to me.

 

[thomate1]

,

Your attempt at a solution is correct. The wavelength of the string would indeed be 60 cm. This can be determined by using the formula Lambda = 2L/n, where L is the length of the string and n is the number of segments in which the string vibrates. In this case, the string vibrates in four segments, so n=4. The length of the string is given as 120 cm, so the wavelength would be 60 cm.

To check if your answer is correct, we can also use the formula for frequency, f = v/lambda, where v is the speed of the wave. In this case, the wave is traveling along the string, so we can use the speed of the wave on a string, which is given by v = sqrt(T/m), where T is the tension in the string and m is the mass per unit length of the string. Since these values are not given, we can assume them to be constant for simplicity. Therefore, the frequency of the wave would be f = v/lambda = sqrt(T/m)/lambda. Since both T and m are constants, the frequency would be directly proportional to the inverse of the wavelength. This means that if the wavelength is halved, the frequency would double and vice versa. In our case, the wavelength has indeed halved from the original 120 cm to 60 cm, so the frequency must have doubled from the original 60 Hz to 120 Hz. This confirms that your answer for the wavelength is correct.

(b) To determine the fundamental frequency of the string, we can use the formula f = nv/2L, where n is the number of segments in which the string vibrates, v is the speed of the wave, and L is the length of the string. In this case, n=4, v is the same as before, and L=120 cm. Plugging in these values, we get f = (4*sqrt(T/m))/2(120 cm) = sqrt(T/m)/60 cm. Since T and m are constants, we can see that the fundamental frequency is directly proportional to the inverse of the length of the string. This means that if the length of the string is halved, the fundamental frequency would double and vice versa. In our case, the length of the string has been halved from the original 240 cm to 120 cm, so the fundamental frequency would double from the original 60 Hz