A phone cord is 4.75 m long. The cord has a mass of 0.200 kg. A transverse wave pulse is produced by plucking one end of the taut cord. The pulse makes four trips down and back along the cord in 0.900 s. What is the tension in the cord?
The speed of the wave is 8*4.75 m/0.9 s
= 42.2 m/s
The lineal density of the cord is
s = 0.042 kg/m
The wave speed is sqrt(T/s)= 42.2
Solve for the tension, T
To find the tension in the cord, we can use the wave speed formula:
Wave Speed (v) = Frequency (f) × Wavelength (λ)
In this case, the distance traveled by the pulse is equal to four times the length of the phone cord, so the wavelength is 4 × 4.75 m = 19 m.
The frequency can be calculated by dividing the number of trips the pulse makes by the time taken:
Frequency (f) = Number of Trips / Time = 4 trips / 0.900 s = 4.444 Hz
Now we can calculate the wave speed:
Wave Speed (v) = Frequency (f) × Wavelength (λ) = 4.444 Hz × 19 m = 84.436 m/s
The wave speed can also be determined from the tension in the cord and the linear mass density (μ) using the formula:
Wave Speed (v) = √(Tension (T) / Linear Mass Density (μ))
Now, we need to find the linear mass density (μ) of the cord. The linear mass density can be calculated by dividing the mass of the cord by its length:
Linear Mass Density (μ) = Mass (m) / Length (L) = 0.200 kg / 4.75 m = 0.04211 kg/m
By substituting the values in the wave speed formula, we can solve for the tension (T):
84.436 m/s = √(T / 0.04211 kg/m)
Squaring both sides of the equation to isolate the tension:
(84.436 m/s)^2 = T / 0.04211 kg/m
7080.02 m^2/s^2 = T / 0.04211 kg/m
T = 7080.02 m^2/s^2 × 0.04211 kg/m
T ≈ 298.37 N
Therefore, the tension in the cord is approximately 298.37 Newtons.