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

AI Thread Summary
The discussion revolves around calculating the mass of a section of string supporting a sphere using principles of standing waves and tension. The fundamental frequency is given as 246 Hz, with the string length determined through trigonometry to be 0.789 m. The tension was initially calculated using T = mg/sinθ, but it was later clarified that T should simply be mg since the string passes over a smooth rod. The derived equation for mass, m = (TL/[fλ]^2), resulted in a value of 0.000927 kg, which was marked incorrect by the CAPA system. Participants expressed confidence in their methods but questioned the accuracy of the CAPA feedback.
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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.
 

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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.
 
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