Question

Just before launching from earth, an astronaut ties a small mass m to the ceiling of his cockpit using a string of length L and mass per unit length μ. The mass of the string is significantly smaller than the mass of the tied object. He then plucks the string and measures the frequency of its n = 1 and n = 2 standing waves. As his rocket takes off, with an acceleration of ar, he repeats the measurement. In terms of m,g,ar,L, and μ,

1. derive an expression for the tension in the string before and during takeoff;
2. derive an expression for the speed of a wave on the string before and during takeoff;
3. find the frequency of the n = 1 and n = 2 standing-wave modes before and during takeoff.
4. What acceleration must the rocket have so that the frequency of the n = 2 mode before the launch is equal to the n = 1 mode during takeoff?

Write down the equation for the speed of a wave on a string. Express your answer in terms of some or all the variables T, tension on the string, μ, g, and L.

Write down the equation for the frequency of a standing wave on a string. Express your answer in terms of some or all the variables v, speed of a wave on a string, g, n, the number of the harmonic, ar, L, and μ.

Answer 1

when rocket is at rest, velocity v_o of standing wave is given by

.........................(1)

where T = mg , is tension due to mass hanging at the end of string ,

is mass of string per unit length .

when rocket is moving fast with acceleration a_r , then the tension T = m( g+a_r )

Hence when rocket is moving with acceleration a_r , velocity v_a of standing wave is given by,

.............................(2)

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for n = 1 standing wave, number of anti-node is 1 , hence wavelength and length l of string are related as

l = ( / 2 )   or = 2l

Hence frequency f for n = 1 standing wave is obtained from the relation , f = v /

frequency f_o for n = 1 standing wave, when rocket is at rest is given by

.............................(3)

frequency f_a for n = 1 standing wave, when rocket is moving with acceleration a_r is given by

............................(4)

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for n = 2 standing wave, number of anti-node is 2 , hence wavelength and length l of string are related as

l =       or = l

frequency f_o for n = 2 standing wave, when rocket is at rest is given by

...............................(5)

frequency f_a for n = 2 standing wave, when rocket is moving with acceleration a_r is given by

.........................(6)

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Frequency of n=2 mode before launch is given by eqn.(5)

Frequency of n=1 mode during take-off is given by eqn.(4)

Required acceleration to equalise the above frequencies is obtained by equating eqn.(5) and (4)

By simplifying above eqn., we get a_r = 3g