A standing wave on a string with a mass hanging on it

[paul11]

Homework Statement

A sphere of mass M=6.85 kg is supported by a string that passes over a light horizontal rod of length L=0.73 m.

Given that the angle is θ=22.3° and that the fundamental frequency of standing waves in the section of the string above the horizontal rod is f=246 Hz, determine the mass of this section of the string.

Homework Equations

v = √(T/μ)
μ = m / L
v = fλ

The Attempt at a Solution

First off, I found the length of the string by using trig since it's a right triangle. L = 0.789 m (different L than on the attached picture)

It is the fundamental frequency so the λ = 2L. λ = 1.57 8m

The tension I found through looking at its FBD from which I obtained T = mg/sinθ. T = 177.09 N

Now I combined eliminating v from two of the equations above, I derived fλ = √(T/μ). Then I substitute μ = m / L and isolate m. I get an equation of m = (TL/[fλ]^2)

Using the derived equation, I calculate a value of m = 0.000927 kg, but the CAPA says it is wrong. I was quite confident on my approach, to the point that I'm doubting the CAPA now, so I don't know where my errors are.

 

[Stonebridge]

The tension in the string is not mg/sinθ
it's just mg as the string passes over a (I presume) smooth end of the rod.
Otherwise the method is correct.