A phone cord is 3.26 m long. The cord has
a mass of 0.224 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 0.865 s.
What is the tension in the cord?
Answer in units of N
The wave speed is
V = 8*3.26/0.865 = 30.2 m/s.
Cord mass per unit length is
sigma = 0.0687 kg/m
Wave speed is
V = sqrt[T/sigma]
Solve for T
It will be in Newtons
To find the tension in the cord, we can use the formula for the wave speed:
v = λ * f
where v is the wave speed, λ is the wavelength, and f is the frequency.
We also know that the wave speed is equal to the distance traveled by the pulse divided by the time taken:
v = 2 * d / t
where d is the distance traveled by the pulse and t is the time taken.
In this case, the pulse makes four trips down and back, so the distance traveled by the pulse is 4 times the length of the cord:
d = 4 * 3.26 m = 13.04 m
Substituting this into the equation for the wave speed:
v = 2 * 13.04 m / 0.865 s
v = 30.08 m/s
Now, we need to find the frequency of the wave. The frequency is given by:
f = 1 / T
where T is the period. The period is the time taken for one complete cycle of the wave.
In this case, the wave makes four trips down and back in 0.865 s, so the time taken for one trip down and back is:
T = 0.865 s / 4
T = 0.21625 s
Substituting this into the equation for frequency:
f = 1 / 0.21625 s
f = 4.621 Hz
Now we can substitute the wave speed and frequency into the equation for tension:
T = (μ * v^2) / λ
where μ is the linear mass density of the cord (mass per unit length) and λ is the wavelength.
The linear mass density is given by:
μ = m / L
where m is the mass of the cord and L is the length of the cord.
Substituting the given values:
μ = 0.224 kg / 3.26 m
μ = 0.068711 kg/m
Now we need to find the wavelength. The wavelength is given by:
λ = v / f
Substituting the values we found:
λ = 30.08 m/s / 4.621 Hz
λ = 6.508 m
Finally, substituting the values into the equation for tension:
T = (0.068711 kg/m * (30.08 m/s)^2) / 6.508 m
T ≈ 79.3 N
Therefore, the tension in the cord is approximately 79.3 N.
To find the tension in the cord, we can use the wave speed equation which states that the wave speed (v) is equal to the frequency (f) multiplied by the wavelength (λ).
In this case, the pulse makes four trips down and back along the cord. A round trip would be equivalent to two wavelengths, so the number of wavelengths (n) can be calculated by dividing the number of trips by 2: n = 4/2 = 2.
The wavelength can then be found by dividing the length of the cord by the number of wavelengths: λ = 3.26 m / 2 = 1.63 m.
Next, we can calculate the wave speed by dividing the wavelength by the time it takes for the pulse to make one round trip: v = λ / t = 1.63 m / 0.865 s = 1.886 m/s.
The wave speed is also related to the tension (T) and the mass per unit length (μ) of the cord by the equation: v = √(T / μ).
Rearranging this equation to solve for T:
T = v^2 * μ
To find μ, we divide the mass of the cord by its length: μ = mass / length = 0.224 kg / 3.26 m = 0.0687 kg/m.
Finally, we can substitute the values into the equation to find the tension:
T = (1.886 m/s)^2 * 0.0687 kg/m = 0.231 N.
Therefore, the tension in the cord is approximately 0.231 N.