Question

Fellow student of mathematical bent tells you that the wave function of a traveling wave on a thin rope is y(x,t)= 2.20 mm cos[( 7.02 rad/m )x+( 743 rad/s )t]. Being more practical-minded, you measure the rope to have a length of 1.33 m and a mass of 3.31 g .
A) AmplitudeB) FrequencyC) WavelengthD) Wave SpeedE) Direction the wave is travelingF) Tension in the ropeG) Average power transmitted by the wave

Answer 1

Answer

given,

y(x,t)= 2.20 mm cos[( 7.02 rad/m )x+( 743 rad/s )t]

length of the rope = 1.33 m

mass of the rope = 3.31 g

comparing the given equation from the general wave equation

y(x,t)= A cos[k x+ω t]

A is amplitude

now on comparing

a) Amplitude  = 2.20 mm

b) frequency =

     $f = \dfrac{\omega}{2\pi}$

     $f = \dfrac{743}{2\pi}$

          f = 118.25 Hz

c) wavelength

        $k= \dfrac{2\pi}{\lambda}$

        $\lambda= \dfrac{2\pi}{k}$

        $\lambda= \dfrac{2\pi}{7.02}$

        $\lambda= 0.895\ m$

d) speed

         $v = \dfrac{\omega}{k}$

         $v = \dfrac{743}{7.02}$

                v = 105.84 m/s

e) direction of the motion will be in negative x-direction

f) tension

  $T = \dfrac{v^2\ m}{L}$

  $T = \dfrac{(105.84)^2\times 3.31 \times 10^{-3}}{1.33}$

      T = 27.87 N

g) Power transmitted by the wave

  $P = \dfrac{1}{2}m\ v \omega^2\ A^2$

  $P = \dfrac{1}{2}\times 0.00331\times 105.84\times 743^2\ 0.0022^2$

      P = 0.438 W