In an acoustics experiment, a piano string with a mass of 5.03g and a length of

Question

In an acoustics experiment, a piano string with a mass of 5.03g and a length of 70.7cm is held under tension by running the string over a frictionless pulley and hanging a 251kg weight from it. The whole system is placed in an elevator.


a) What is the fundamental frequency of oscillation for the string when the elevator is at rest?

b) With what acceleration and in what direction (up or down) should the elevator move for the string to produce the proper frequency of 440.Hz, corresponding to middle A?

In an acoustics experiment, a piano string with a mass of 5.03g and a length of 70.7cm is held under tension by running the string over a frictionless pulley and hanging a 251kg weight from it. The whole system is placed in an elevator.


a) What is the fundamental frequency of oscillation for the string when the elevator is at rest?

b) With what acceleration and in what direction (up or down) should the elevator move for the string to produce the proper frequency of 440.Hz, corresponding to middle A?

Explanation / Answer

v = sqrt(T/mu)

a) when at rest T = m g

f = v/2L = sqrt( m g/mu)/(2 L) = sqrt(251*9.81/(5.03E-3/0.707))/(2*0.707)= 416 Hz

b) now

T - m g = m a

T = m ( g + a)

so

440 = sqrt(251*(9.81+a)/(5.03E-3/0.707))/(2*0.707)

a=1.16 m/s^2 upwards

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