Question

A string of length 1.3 m is oscillating in a standing wave pattern. If the tension in the string is 430 N, the string has a mass of 23 g/m, and the amplitude of the oscillations is 2.1 mm, what is the maximum speed of a point on the string when it is oscillating in the fundamental mode?
A

1.4 m/s

B

0.69 m/s

C

0.45 m/s

D

0.22 m/s

E

2.8 m/s

Answer 1

To solve this problem we will use the concepts related to the speed of a string which is given by the applied voltage and the linear mass density of it. With the speed value we can find the fundamental frequency that will serve as a step to find the maximum speed through the relation of Amplitude and Angular Speed. So:

$v = \sqrt{\frac{T}{\mu_e}}$

Where,

T = Tension

$\mu_e$= Linear mass density

$v = \sqrt{\frac{430}{0.023}}$

$v = 136.7m/s$

With this value the fundamental frequency would be

$f = \frac{v}{2L}$

$f = \frac{136.7}{2*1.3}$

$f = 52.6Hz$

Finally the maximum speed is given with the relation between the Amplitude (A) and the Angular frequency, then

$V_{max} = A\omega$

$V_{max} = A(2\pi f)$

$V_{max} = (2.1*10^{-3})(2\pi 52.6)$

$V_{max} = 0.69m/s$

Therefore the correct answer is B.