Astronauts visiting Planet X have a 2.6m -long string whose mass is 4.9g . They

Question

Astronauts visiting Planet X have a 2.6m -long string whose mass is 4.9g . They tie the string to a support, stretch it horizontally over a pulley 2.0m away, and hang a 1.5kg 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 64hz and 80hz , but at no frequencies in between. What is the value of g , the free-fall acceleration, on Planet X? please use these numbers, not similar numbers from a different problem. thank you!

Explanation / Answer

We know that the velocity of a wave pulse in a string is given by the square root of the tension in the string divided by the linear mass density [sqrt(F/u)]
We also know that the linear mass density (u) can be found by dividing the mass of the string by its length [u = m/L]

The formula for standing wave harmonics is [ f = (nv)/(2L) ]
(I'd like to note that this L is the length 1.8m, because those are the fixed ends, wheras the L that you use for the linear mass density will be the total length of the string (2.6m))

Now the only additional piece of information that we need is the harmonic number (n). To get this we can use the two frequencies, which we know must be only 1 harmonic apart (because there were no frequencies in between which created a standing wave). So n2 = n1 + 1. We also know from the formula above that v and L will not change, so the ratio f/n will be CONSTANT. This gives us f1/n1 = f2/n2.

so (rearranging) we know that the ratios n2/n1 = f2/f1, but we also know that n2 = n1 + 1
so we can find n1 = 1 / [ (f2/f1) - 1 ]

Finally, combining all of these equations, we get an equation

g = (Ms / (Mw*Ls))*[ (2*Ln*f1) * [(f2/f1) - 1] ] ^2
*Where Ms is the mass of the entire string (5.20g = 0.00520 kg), Mw is the mass of the free weight (1.10 kg), Ls is the length of the entire string (2.60m), Ln is the length of the standing wave region (1.8m), f1 is the lowest frequency (64 Hz), and f2 is the higher frequency (80 Hz).

g = ((0.00520kg) / ((1.10kg)*(2.60m)) * [ (2 * (1.80m)*(64Hz) * [(80Hz/64Hz) -1 ] ]^2

I get
g = 6.03 m/s^2

If you wanted to know the other pieces of information, I got
u = 0.002 kg/m
n1 = 4
n2 = 5

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