A string with a linear density of 0.035 kg/m and a mass of 0.014 kg is clamped at both ends. Under what tension in the string will it have a fundamental frequency of 100 Hz?

The correct answer is 270N but I don't know how to get there.

Here are some of the equations we can use but if you know of others feel free to mention them.
T=1/f
f=1/T
v=(square root)F/(m/L)
Fn=nv/2L

Thanks for the help and please indicate which equation you chose if you end up picking one of the ones I have above.

The final answer is 270 N not 224N,how come?

Well, let me help you solve this problem with a touch of humor, because who doesn't like a laughing string?

Let's start with the equation v=(square root)(F/(m/L)), which relates the velocity of the wave on the string (v) to the tension in the string (F), the mass of the string (m), and the length of the string (L).

Now, we know that the fundamental frequency (f) of a standing wave on a string is related to its velocity and length by the equation f = v/(2L).

Since we're given the fundamental frequency (100 Hz), we can substitute this into the equation to get: 100 Hz = v/(2L).

Rearranging the equation, we have v = 2L * 100 Hz.

Now, substituting this expression for v into the first equation, we get: 2L * 100 Hz = (square root)(F/(m/L)).

Squaring both sides of the equation to get rid of the square root, we have 4L^2 * 10000 Hz^2 = F/(m/L).

Rearranging the equation, we have F = 4L^2 * 10000 Hz^2 * (m/L).

Canceling out the length (L) terms, we get F = 4 * 10000 Hz^2 * m * L.

Plugging in the given values, we have F = 4 * 10000 Hz^2 * 0.014 kg * 1 m.

Simplifying the expression, we get F = 5600 N.

Ah, but the question asks for tension, not force! Remember that tension (T) is the same as the force applied to the string. Therefore, the tension in the string is 5600 N.

Oh, wait! I forgot to tell you, the humor in this explanation is free of charge! So, don't worry, no extra tension here.

To find the tension in the string, we can start by using the wave speed equation and the fundamental frequency equation. Since the string is clamped at both ends, it will have a fundamental frequency, which is the lowest resonant frequency.

The wave speed equation is given by: v = √(T/μ), where v is the wave speed, T is the tension, and μ is the linear density of the string.

The fundamental frequency equation is: f = v/λ, where f is the fundamental frequency, v is the wave speed, and λ is the wavelength.

Since we are given the fundamental frequency (f = 100 Hz), we can solve the fundamental frequency equation for v: v = f * λ.

Now, let's substitute the value of v into the wave speed equation: f * λ = √(T/μ).

Since the string is clamped at both ends, the wavelength of the fundamental mode is twice the length of the string: λ = 2L. Substituting this in, we have: f * 2L = √(T/μ).

Now, let's solve for T by squaring both sides of the equation: (f * 2L)^2 = T/μ.

Rearranging, we get: T = (f * 2L)^2 * μ.

Substituting the given values: T = (100 Hz * 2 * L)^2 * 0.035 kg/m.

Simplifying, we have: T = (200 * L)^2 * 0.035 kg/m.

Finally, we can calculate the tension using the given mass of the string (m = 0.014 kg) since linear density (μ) is given by μ = m/L: T = (200 * L)^2 * m/L.

Simplifying further, we get: T = 40000 * L * m.

Since we know the value of T is 270 N, we can set up the equation: 270 N = 40000 * L * m.

Now, we can solve for L: L = 270 N / (40000 * m).

Plugging in the given mass (m = 0.014 kg), we get: L = 270 N / (40000 * 0.014 kg).

Evaluating this expression would give us the length of the string (L), which should be used to find the tension using the equation T = 40000 * L * m.

freq*lambda= v=sqrt(F/linearmassdensity)

lambda= 2*length= 2* mass/(massdenstiy)
= .028/(.035) meters=.8m

solve for tension F.

100*.8=sqrt(F/.035)

6400*.035=F=224N

recheck my work to see if there is an error, I don't see it.