Question

A 200 Hz wave with amplitude 1 mm travels on a long string of linear mass density $6g{m}^{-1}$ kept under a tension of 60 N. (a) Find the average power transmitted across a given point on the string. (b) Find the total energy associated with the wave in a 2.0 m long portion of the string.

Answer 1

$A=1mm={10}^{-3}m,$

$m=6g/m=6\times {10}^{-3}kg/m$

T = 60 N, f = 200 Hz

$V=\sqrt{\left(\frac{T}{m}\right)}=\sqrt{\frac{\left(60\right)}{(6\times {10}^{-3})}}={10}^2=100m/s$

(a) $P_{average}=2{\pi }^{2}mv{A}^2{f}^2$

$=2\times (3.14{)}^2\times (6\times {10}^{-3})\times 100\times ({10}^{-3}\times {200}^2)$

$=473\times {10}^{-3}=0.47W$

(b) Length of the string is 2m.

So, $t=\frac{2}{100}=0.02sec$

Energy $=2{\pi }^{2}mv{f}^2{A}^{2}t$

$=2\times (3.14{)}^2\times 6\times ({10}^{-3})\times 100\times {200}^2\times ({10}^{-3}{)}^2\times \left(0.02\right)$

$=946.5\times {10}^{-5}$

$=9.46\times {10}^{-3}J=9.46mJ$