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

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 = (\omega)/(2\pi)`

`f = (743)/(2\pi)`

f = 118.25 Hz

c) wavelength

`k= (2\pi)/(\lambda)`

`\lambda= (2\pi)/(k)`

`\lambda= (2\pi)/(7.02)`

`\lambda= 0.895\ m`

d) speed

`v = (\omega)/(k)`

`v = (743)/(7.02)`

v = 105.84 m/s

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

f) tension

`T = (v^2\ m)/(L)`

`T = ((105.84)^2* 3.31 * 10^(-3))/(1.33)`

T = 27.87 N

g) Power transmitted by the wave

`P = (1)/(2)m\ v \omega^2\ A^2`

`P = (1)/(2)* 0.00331* 105.84* 743^2\ 0.0022^2`

P = 0.438 W