Question

A tuning fork of frequency 440 Hz is attached to a long string of linear mass density $0.01kg{m}^{-1}$ kept under a tension of 49 N. The fork produces transverse waves of amplitude 0.50 mm on the string. (a) Find the wave speed and the wavelength of the waves. (b) Find the maximum speed and acceleration of a particle of the string. (c) At what average rate is the tuning fork tranmitting energy to the string ?

Answer 1

(a) We know, $v=\sqrt{\left(\frac{T}{m}\right)}$

$=\sqrt{\frac{49}{0.01}}=70m/sec$

And, $v=\frac{n}{\alpha }$

$\therefore \alpha =\frac{v}{n}=\frac{70}{440}=16cm$

(b) We have, y =A sin (wt - kx)

$\therefore \alpha =\frac{v}{n}=\frac{70}{440}=16cm$

$\therefore v_{max}=(\frac{dy}{dt}{)}_{max}=Aw$

$=0.50\times {10}^{-3}\times 2\pi \times 440$

$=1.3816m/sec.$

Again, $a=\frac{d^{2}y}{d{t}^2}$

$=-A{w}^{2}sin(wt-kx)$

$a_{max}=(\frac{d^{2}y}{d{t}^2}{)}_{max}=-A{w}^2$

$=0.50\times {10}^{-3}\times 4{\pi }^2(440{)}^2$

$=3.8km/sec.$

(c) $p=2{\pi }^{2}v{A}^2{n}^2$

$=2\times 10\times 0.01\times 70\times 0.5\times 0.5\times {10}^{-6}\times (440{)}^2$

$=0.67W$