How Do Astronauts Calculate Gravity on Planet X Using Standing Waves?

[momogiri]

Question
Astronauts visiting Planet X have a 2.50m-long string whose mass is 4.60g. They tie the string to a support, stretch it horizontally over a pulley 1.50m away, and hang a 1.70kg 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 acceleration due to gravity, on Planet X?

Attempt
No idea. At all. I tried a bunch of numbers and they're not right XD
Can someone explain how to get it please? :D

 

[Sinan tezok]

Hi Momogiri,

Firstly let's look at a way to approach this problem. We know that we have a 2.50m long string with mass 4.60g. We are told that it is stretched 1.50m and a mass of 1.70 kg is on the free end. We also know that 2 standing waves exist at frequencies 64 Hz and 80 Hz. We know that this string is under a certain tension and that is:

1.70 kg x gravitational pull of planet X

From this it is clear that we must use the formula v² = T/µ which can be rewritten as:

v² = mass * gravitational pull/µ

We can easily find µ since both the length and the mass of the rope are given to us. We also need to make use of another formula to find which node (m) we are dealing with. We know that :

v = 2Lf / m and that the wave speed is the same in both cases therefore:

2Lf₁/ m = 2Lf₂/ (m+1)

After solving for v you can then solve for the gravitational pull of planet X by rearranging v² = mass * gravitational pull/µ. Hope this was helpful,

Cheers,

Sinan

 

[momogiri]

Wow, thanks Sinan! That really helps!
For future reference, the answer was 2.49m/s^2