I don\'t understand how to do this problem. May someone help me through it? Than

Question

I don't understand how to do this problem. May someone help me through it? Thank you.

Astronauts visiting Planet X have a 2.3 m -long string whose mass is 4.8 g. They tie the string to a support, stretch it horizontally over a pulley 1.8 m away, and hang a 1.1 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 free-fall acceleration, on Planet X? Express your answer using two significant figures.

Explanation / Answer

Given that

Astronauts visiting Planet X have a length = 2.3m -long

The string whose mass is (m) = 4.8g =4.8*10-3kg=0.0048kg .

They tie the string to a support, stretch it horizontally over a pulley(d) = 1.8m away,

The mass hang to the free end is (M) =1.1kg

Their data show that standing waves exist at frequencies of 64 Hz and 80 Hz, but at no frequencies in between.

The value of g ,the free-fall acceleration on the planet X is given

The speed (v) =Sqrt(T/u)

Where T =mg =1.1kg*g

The linear density is u =0.0048kg/2.3m=2.086*10-3kg/m

And given that L =1.8m

Now for a string the frequencies are

f =nv/2L ----------(1)

Therefore 64 and (n+1)v/2L =80----(2)

Now dividing equation (1) and (2) we get

(n+1)/n =80/64 =1.25

1.25n =n+1

or n =4

Now substituting the value n=4 in equation (1) we get

4v/2L =64 ==> v =32*L =32*1.8=57.6m/s

Therefore

57.6 =Sqrt(1.1kg*g/2.086*10-3kg/m)

g =(57.6)2*2.086*10-3kg/m/1.1=6.2916m/s2 =6.3m/s2

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