Beat frequency: which equation to use to find the mass of string 2

[Bri15]

Homework Statement

Two strings of equal length, L, are each clamped at both ends and are under the same
tension T. String 1 has a mass of 45.00 g and vibrates at a frequency of 512 Hz. String 2 is
slightly heavier. When the two strings are plucked at the same time, beats with a
frequency of 4.0 Hz are heard.

(i) What is the vibrational frequency of string 2?

(ii) What is the mass of string 2?

Homework Equations

Fbeat=f1-f2
f= sqrt (T/(m/L))/2L

The Attempt at a Solution

f2=508Hz (a)
(b) I do not understand which equation is applicable to find the mass

 

[mark726]

of string 2. Could you please provide more information about the setup and any other known variables?

Without any additional information, it is difficult to determine the mass of string 2. However, we can make some assumptions and use the given information to estimate the mass.

Assuming that the two strings have the same linear density (mass per unit length), we can use the equation for the frequency of a vibrating string to find the mass of string 2:

f = sqrt(T/(m/L))/2L

Where:
f = frequency of vibration
T = tension in the string
m = mass of the string
L = length of the string

We know that the tension (T) and length (L) are the same for both strings. We also know the frequency (f) of string 1 (512 Hz) and the frequency of the beats (4 Hz).

Using these values, we can set up the following equations:

512 Hz = sqrt(T/(m/L))/2L for string 1
508 Hz = sqrt(T/(m/L))/2L for string 2

Solving for T/(m/L) in both equations, we get:

T/(m/L) = (512 Hz * 2L)^2 = 524,288 Hz^2 * L^2 for string 1
T/(m/L) = (508 Hz * 2L)^2 = 515,584 Hz^2 * L^2 for string 2

Since we know that the tension (T) and length (L) are the same for both strings, we can set these two equations equal to each other and solve for the mass of string 2:

524,288 Hz^2 * L^2 = 515,584 Hz^2 * L^2

Dividing both sides by L^2, we get:

524,288 Hz^2 = 515,584 Hz^2

Taking the square root of both sides, we get:

724 Hz = 718 Hz

Since this is not possible, we can conclude that our initial assumption (that the two strings have the same linear density) is not correct. This means that we cannot accurately determine the mass of string 2 without more information about the setup.