A string of mass 0.2kg/m has length l = 0.6m. It is fixed at both ends and stretched such that it has a tension of segments with amplitude =0.5cm.Find frequency of the wave?
The speed of the wave is given by:
v = √(T/μ)
where T is the tension in the string and μ is the linear density (mass per unit length).
Tension of segments with amplitude 0.5cm can be estimated as:
T'=F*0.005
where F is the force required to stretch a segment with amplitude 0.5cm.
For a string fixed at both ends and vibrating in its fundamental mode (i.e. with one antinode in the middle), the wavelength λ is twice the length of the string, so λ = 2l.
Therefore, the frequency f of the wave is given by:
f = v/λ = v/(2l)
Substituting the expressions for v and λ, we get:
f = √(T'/μ)/(2l) = √(F/μ*0.005)/(2l)
We need some additional information about the force required to stretch a segment with amplitude 0.5cm, or about the linear density of the string, in order to calculate the frequency.
To find the frequency of the wave on a string, we can use the formula:
f = v/λ
where f is the frequency, v is the velocity of the wave, and λ (lambda) is the wavelength.
In this case, we are given the tension of the string (T), the mass per unit length (μ), and the amplitude (A).
The velocity of the wave (v) can be determined using the formula:
v = sqrt(T/μ)
To find the wavelength (λ), we can use the formula:
λ = 2A
Given that the amplitude (A) is 0.5 cm, we convert it to meters:
A = 0.5 cm = 0.005 m
Now, let's calculate the velocity (v) using the given tension (T) and mass per unit length (μ):
μ = m/l
where m is the mass and l is the length of the string.
Given that the mass (m) is 0.2 kg/m and the length (l) is 0.6 m:
μ = 0.2 kg/m / 0.6 m = 0.333 kg/m
Now, we can calculate the velocity (v):
v = sqrt(T/μ)
Substituting the values:
v = sqrt(T/(0.333 kg/m))
Finally, let's substitute the values of λ and v into the formula for frequency:
f = v/λ
Substituting the values:
f = (sqrt(T/(0.333 kg/m))) / (2 * 0.005 m)