The mass of a string is 2.5 × 10-3 kg, and it is stretched so that the tension in it is 250 N. A transverse wave traveling on this string has a frequency of 300 Hz and a wavelength of 0.45 m. What is the length of the string?
Mass per unit length m₀ = m/L,
Velocity in the stretched string is
v = sqrt(T/m₀) =sqrt(T•L/m),
λ=v/f = sqrt(T•L/m)/f,
L=f²λ²m/T
To find the length of the string, we need to use the formula for the speed of a wave on a string:
v = √(F/μ),
where v is the speed of the wave, F is the tension in the string, and μ is the linear mass density of the string.
The linear mass density (μ) of the string is given by:
μ = m/ℓ,
where m is the mass of the string and ℓ is the length of the string.
In this case, the mass of the string is given as 2.5 × 10^(-3) kg.
The tension in the string is given as 250 N.
The frequency of the wave is given as 300 Hz.
The wavelength of the wave is given as 0.45 m.
First, we can find the speed of the wave using the formula:
v = fλ,
where f is the frequency of the wave and λ is the wavelength.
Substituting the given values, we get:
v = 300 Hz × 0.45 m = 135 m/s.
Now, we can use the formula for speed to find the linear mass density:
v = √(F/μ).
Rearranging the formula, we get:
μ = F/v^2.
Substituting the given values, we get:
μ = 250 N / (135 m/s)^2.
Calculating this, we get:
μ = 250 N / 18225 m^2/s^2 ≈ 0.0137 kg/m.
Finally, we can use the linear mass density to find the length of the string:
μ = m/ℓ,
rearranging the formula, we get:
ℓ = m/μ.
Substituting the given mass of the string and the calculated linear mass density, we get:
ℓ = (2.5 × 10^(-3) kg) / 0.0137 kg/m.
Calculating this, we get:
ℓ ≈ 0.183 m.
Therefore, the length of the string is approximately 0.183 meters.