When driven by a 120 Hz vibrator, a string has transverse waves of 31 cm wavelength traveling along it. (a) What is the speed of the waves on the string& (b) Is the tension in the string is 1.2 N, what is the mass of 50 cm of the string?
To find the speed of the waves on the string, we can use the equation:
v = f * λ
Where:
v = speed of the wave
f = frequency of the vibrator
λ = wavelength of the wave
(a) To find the speed of the waves on the string, we know the frequency of the vibrator (f = 120 Hz) and the wavelength of the wave (λ = 31 cm). Let's convert the wavelength to meters:
λ = 31 cm = 0.31 m
Now, we can calculate the speed of the waves on the string:
v = 120 Hz * 0.31 m = 37.2 m/s
So, the speed of the waves on the string is 37.2 m/s.
(b) To find the mass of 50 cm of the string, we need to use the tension in the string and the speed of the waves. The equation that relates these variables is:
v = √(T/μ)
Where:
v = speed of the wave
T = tension in the string
μ = linear mass density of the string (mass per unit length)
Rearranging the equation, we can solve for μ:
μ = T / v^2
Using the given tension (T = 1.2 N) and the speed of the waves (v = 37.2 m/s), we can calculate the linear mass density:
μ = 1.2 N / (37.2 m/s)^2
μ = 1.2 N / 1380.48 m^2/s^2
μ ≈ 8.696 x 10^(-4) kg/m
Now, we can calculate the mass of 50 cm (0.5 m) of the string:
mass = μ * length
mass = 8.696 x 10^(-4) kg/m * 0.5 m
mass ≈ 4.348 x 10^(-4) kg
Therefore, the mass of 50 cm of the string is approximately 4.348 x 10^(-4) kg.