A 123-cm-long, 5.00 g string oscillates in its n = 2 mode with a frequency of 200 Hz and a maximum amplitude of 4.50 mm.
What is the tension in the string?
To find the tension in the string, we can use the equation for the frequency of oscillation of a string:
f = (1/2L) * sqrt(T/μ)
where:
f is the frequency of oscillation,
L is the length of the string,
T is the tension in the string, and
μ is the linear mass density of the string.
In this case, we are given the length of the string (L = 123 cm = 1.23 m), the frequency of oscillation (f = 200 Hz), and the linear mass density:
μ = mass / length.
To find the mass, we need to use the given grams and convert it to kilograms:
mass = 5.00 g = 0.00500 kg.
Now, we can find the mass per unit length (μ):
μ = mass / length = 0.00500 kg / 1.23 m = 0.004065 kg/m.
Substituting the given values into the equation, we can solve for T:
200 Hz = (1/2 * 1.23 m) * sqrt(T / 0.004065 kg/m).
To isolate T, we need to square both sides of the equation:
(200 Hz)^2 = (1/2 * 1.23 m)^2 * (T / 0.004065 kg/m).
Simplifying the equation further, we have:
40000 Hz^2 = (0.615 m)^2 * (T / 0.004065 kg/m).
Now, we can solve for T:
T = 40000 Hz^2 * 0.004065 kg/m / (0.615 m)^2.
Evaluating this expression gives us:
T ≈ 161.47 N (rounded to two decimal places).
Therefore, the tension in the string is approximately 161.47 N.
Please explain what you mean by the n=2 mode. Does it have a standing wave with a node in the middle? If so, the wavelength of the traveling waves that form the standing wave is 123 cm. Get the wave speed c from
c = (wavelength)*(frequency)
You already know the frequency.
The wave speed can be used to solve for the string tension, T
c = sqrt [T/(mass per length)]
= sqrt(T*L/m)
The amplitude does not affect the answer.
I suggest using mass in g and length in cm. The answer (T) will then be in dynes.
10^7 dynes = 1 Newton