A 121 cm-long, 3 g string oscillates in its m = 3 mode with a frequency of 154 Hz and a maximum amplitude of 5.0 mm. What are the wavelength and the tension in the string?

1.21 m long so wavelength = 2L/3 = .807m

.003 kg/1.21 m = .00248 kg/m
v = sqrt(Tension/kg/m) =sqrt(T/.00248)
f = n v/(2L)
154 = 3 v/(2.42)
v = 124.2m/s
124.2 = sqrt(T/.00248)
T/.00248 = 15425
T = 38.3 Newtons

The amplitude has nothing to do with it.

Thank you so very much Damon!

To find the wavelength and the tension in the string, we can use the formulas for the frequency and the wave speed of the string.

The frequency of the string can be calculated using the formula:

f = (n/2L) * sqrt(T/μ)

Where:
- f is the frequency of the string (154 Hz in this case)
- n is the mode number (m in this case)
- L is the length of the string (121 cm, which is 1.21 m)
- T is the tension in the string (we'll denote it as Tt to avoid confusion with the tension we're trying to find)
- μ is the linear mass density of the string

We can rearrange the formula to solve for T:

T = (μ * (f * 2L)^2) / n^2

Now, let's calculate μ, the linear mass density of the string. The linear mass density is defined as the mass per unit length, so we can find it by dividing the mass of the string by its length:

μ = m / L

Substituting the given values:
m = 3 g = 0.003 kg
L = 1.21 m

μ = 0.003 kg / 1.21 m

Now, let's substitute the values into the formula for T and calculate it:

T = (0.003 kg / 1.21 m) * ((154 Hz * 2 * 1.21 m)^2) / (3^2)

Simplifying the equation, we find the value of T:

T ≈ 167.66 N

Now, let's calculate the wavelength. The wave speed of the string can be found using the formula:

v = f * λ

Where:
- v is the wave speed of the string
- λ is the wavelength of the string (we're trying to find it)

We can rearrange the formula to solve for λ:

λ = v / f

The wave speed of the string can be found using the formula:

v = sqrt(T / μ)

Let's substitute the values into the formula for v and calculate it:

v = sqrt(167.66 N / (0.003 kg / 1.21 m))

Now, let's substitute the value of v and f into the formula for λ and calculate it:

λ = sqrt(167.66 N / (0.003 kg / 1.21 m)) / 154 Hz

Simplifying the equation, we find the value of λ:

λ ≈ 0.825 m

Therefore, the wavelength of the string is approximately 0.825 meters (or 82.5 cm) and the tension in the string is approximately 167.66 N.