The mass of a string is 6.8 x 10-3 kg, and it is stretched so that the tension in it is 210 N. A transverse wave traveling on this string has a frequency of 190 Hz and a wavelength of 0.67 m. What is the length of the string?
v = sqrt(T/m₀) =sqrt(TL/m)
λ =v/f => v= λf
(λf)²=TL/m
L=(λf)²m/T
thank you
0.52
To find the length of the string, we need to use the equation for wave speed.
The wave speed (v) is given by the equation:
v = f * λ
where:
v = wave speed in meters per second (m/s)
f = frequency in hertz (Hz)
λ = wavelength in meters (m)
We know the frequency (f) is 190 Hz and the wavelength (λ) is 0.67 m.
Plugging these values into the equation, we get:
v = 190 Hz * 0.67 m = 127.3 m/s
Now, to find the length of the string, we can use the wave speed equation as follows:
v = √(T / μ)
where:
v = wave speed in meters per second (m/s)
T = tension in newtons (N)
μ = mass per unit length in kilograms per meter (kg/m)
We are given the tension (T) as 210 N.
To find μ (mass per unit length), we can divide the mass (m) by the length (L) of the string:
μ = m / L
We are given the mass (m) as 6.8 x 10^(-3) kg.
Rearranging the equation, we get:
L = m / μ
Substituting the given values into the equation, we have:
L = (6.8 x 10^(-3) kg) / μ
Now, let's find μ by substituting the given values into the wave speed equation:
v = √(T / μ)
Rearranging the equation, we have:
μ = T / (v^2)
Substituting the given values, we get:
μ = 210 N / (127.3 m/s)^2 = 210 N / 16198.29 m^2/s^2 = 0.01296 kg/m
Now, substituting the calculated value of μ into the equation for string length:
L = (6.8 x 10^(-3) kg) / (0.01296 kg/m)
Calculating, we have:
L = 524.69 m
Therefore, the length of the string is approximately 524.69 meters.