Finding gravity on Planet X from a string, a weight, and frequency

[LBRRIT2390]

Homework Statement

Astronauts visiting Planet X have a 2.5 m-long string whose mass is 5.0 g. They tie the string to a support, stretch it horizontally over a pulley 1.8 m away, and hang a 1.3 kg mass on the free end. Then the astronauts begin to excite standing waves on the string. Their data show that standing waves exist at frequencies of 64 Hz and 80 Hz, but at no frequencies in between.

What is the value of g, the acceleration due to gravity, on Planet X?

Homework Equations

Fundamental frequency of a stretched string
f₁=$$\frac{1}{2L}$$$$\sqrt{\frac{T_s}{\mu}}$$

Linear Density of a string
$$\mu$$ = $$\frac{mass}{length}$$

The Attempt at a Solution

T_s = 1.3kg(g)

$$\mu$$ = $$\frac{0.005kg}{2.5m}$$ = 0.002

L = 1.8m

f₁ = 64Hz

Place in Fundamental frequency equation and solve for g.


64Hz=$$\frac{1}{2(1.8m)}$$$$\sqrt{\frac{1.3*g}{0.002}$$

for f₁ = 64Hz, I got g = 81

I tried substituting different frequencies and using a different L but I could not arrive at the correct answer which is g=5 m/s²


Can someone please teach me how to arrive at this correct answer?

 

[Choppy]

Here's a couple of hints.

You're given two data points in the question - that standing waves exist at the two given frequencies. So it's likely that both of them are required to arrive at the answer.

It's tempting to assume that the given data occur at the first and second natural modes of vibration, but from the wording of the question, all we can really say for sure is that they exist at the nth and (n+1)th modes - since it's given that no standing waves were found to exist at frequencies in between. Hence, we're not necessarily dealing with the fundamental frequency (n = 1) case.

Hope this helps.

 

[LBRRIT2390]

I am still lost...