To solve this problem, we can use the equation that relates the speed of a wave on a string to its frequency and wavelength:
v = fλ
where: v is the speed of the wave, f is the frequency of the wave, and λ is the wavelength of the wave.
We can also use the equation that relates the speed of the wave on a string to the tension (T) and mass per unit length (μ) of the string:
v = √(T/μ)
where: T is the tension in the string, and μ is the mass per unit length of the string.
First, we can calculate the speed of the wave using the given frequency and wavelength:
v = fλ = 300 Hz * 0.63 m = 189 m/s
Now, we can use the speed of the wave, the tension, and the mass per unit length to find the length of the string:
v = √(T/μ) 189 m/s = √(270 N / μ)
To proceed, we need to find the mass per unit length (μ) of the string. Mass per unit length is calculated by dividing the total mass of the string by its length (L):
μ = mass / length
Given that the mass of the string is 5×10^-3 kg, we need to find the length (L) of the string. Rearranging the equation, we have:
L = mass / μ L = 5×10^-3 kg / μ
We substitute this value of L into the equation for the speed of the wave:
189 m/s = √(270 N / (5×10^-3 kg / L)) 189 m/s = √(270 N * L / 5×10^-3 kg) 189^2 m^2/s^2 = 270 N * L / 5×10^-3 kg (189^2 m^2/s^2) * (5×10^-3 kg) = 270 N * L (189^2 * 5×10^-3) m^2/s^2 = 270 N * L L = (189^2 * 5×10^-3) m^2/s^2 / 270 N
Evaluating the expression on the right-hand side, we can calculate the length (L) of the string:
L = (189^2 * 5×10^-3) m^2/s^2 / 270 N L ≈ 11.92 m
Therefore, the length of the string is approximately 11.92 meters.