A streched string is forced to transmit transverse waves by means of an oscillator coupled to one end. The string has a diameter of `4mm`. The amplititude of the oscillation is `10^(-4)` m and the frequency is `10Hz`. Tension in the string is `100` N and mass density of wire is `4.2xx10^(3)kg//m^3`. Find
(a) the equation of the waves along the string
(b) the energy per unit volume of the wave
(c) the average energy flow per unit time across any section of the string and
(d) power required to drive the oscillator.

Answer

A, B, C, D

Explanation

(a) Speed of transverse wave on the string is
`v=sqrt((T)/(rhoS))` (as`mu=rhoS`)
Substituting the values,we have
`v=sqrt((100)/((4.2xx10^3)((pi)/(4))(4.0xx10^-3)^2))=43.53m//s`
`omega=2pif=20pi=62.83 rad//s`
`k=(omega)/(v)=1.44m^-1`
:. Equation of the waves along the string, `y(x,t)=A sin (kx-omegat)`
`= (10^-4m) sin[(1.44m^-1)x-(62.83 rade//s)t]`
(b) Energy per unit volume of the string,`u="energy density"=(1)/(2)rhoomega^2A^2`
Subsitutituting the values, we have
`=u=((1)/(2))(4.2xx10^3)(62.83)^2(10^-4)^2`
`= 8.29xx10^-2 J//m^3`
(c) Average energy flow per unit time, `P=power=((1)/(2)rhoomega^(2)A^(2))(Sv)=(u)(Sv)`
Susbstituting the values, we have `P=(8.29xx10^(-2))((pi)/(4))(4xx10^(-3)) ^(2) (43.53)`
`4.53xx10^-5 J//s`
(d) Therefore, power required to drive the oscillator is `4.53xx10^(-5)W`.