A guitar string of length 64 cm and mass 6.16 g is fixed at both ends. The wave generated travels along the string at a speed of 642 ms-1.
(i)What is the frequency of the third harmonic?
(ii)What is the tension of the string?
(iii)Suppose the guitar string is replaced by a thinner string such that the mass per unit length is a quarter of
the original value, calculate the third harmonic frequency after the replacement if the tension remained
the same.
To answer the questions, we need to use the formulas related to wave properties on a string.
(i) To find the frequency of the third harmonic, we need to first determine the fundamental frequency and then calculate the frequency of the third harmonic.
The fundamental frequency (f1) of a fixed string can be calculated using the formula:
f1 = v / λ
where v is the velocity of the wave and λ is the wavelength.
Given:
v = 642 m/s
λ = 2L (for the fundamental frequency)
L is the length of the string, which is 64 cm. Converting cm to meters: L = 64 cm = 0.64 m.
Using the formula, the fundamental frequency (f1) is:
f1 = v / λ = 642 m/s / (2 * 0.64 m) = 500 Hz
Now, to find the frequency of the third harmonic (f3), we can use the relationship:
f3 = 3f1
So, the frequency of the third harmonic is:
f3 = 3 * 500 Hz = 1500 Hz
Therefore, the frequency of the third harmonic is 1500 Hz.
(ii) To find the tension of the string, we can use the formula:
v = √(T / μ)
where v is the velocity of the wave, T is the tension, and μ is the mass per unit length of the string.
Given:
v = 642 m/s
μ = mass / length
The length of the string is given as 64 cm, which is 0.64 m. So, the mass per unit length is:
μ = 6.16 g / 0.64 m = 9.625 g/m
To convert grams to kilograms, divide by 1000:
μ = 9.625 g/m / 1000 = 0.009625 kg/m
Using the formula, we can rearrange it to solve for T:
T = μ * v^2
Calculating the tension (T):
T = 0.009625 kg/m * (642 m/s)^2 = 39.25 N
Therefore, the tension of the string is 39.25 N.
(iii) If the mass per unit length is reduced to a quarter of the original value while maintaining the same tension, we can calculate the new frequency of the third harmonic.
Given that the mass per unit length has decreased to a quarter of the original value, the new mass per unit length (μ') would be:
μ' = μ / 4
Substituting the original value of μ from before:
μ' = 0.009625 kg/m / 4 = 0.00240625 kg/m
Now, using the same tension (T) value as before, we can calculate the new frequency of the third harmonic (f3') using the formula:
f3' = (v / λ')
where λ' is the new wavelength for the third harmonic.
We know that the new wavelength (λ') is equal to 2 times the length of the new string, which is half the length of the original string:
λ' = 2 * (L / 2) = L
Substituting the given length (L) of the original string:
λ' = 0.64 m
Now, substituting the values into the formula, we can calculate f3':
f3' = v / λ' = 642 m/s / 0.64 m = 1000 Hz
Therefore, the frequency of the third harmonic after the replacement is 1000 Hz.