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A travelling harmonic wave on a stringis described by y ( x,t ) = 7.5 sin ( 0.0050x +12 t +pi //4) (a) What are the displacement and velocity of oscillation of a point at x=1cm, and t =1s ? Is this velocity equal to the velocity of wavepropagation ? (b) Locate the points of the string whcih have the same transverse displacements and velocityas the x=1 cmpoint at t=2s, 5s and 11s. |
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Answer» Solution :(a) The travelling harmonicwave is y ( x,t) `= 7.5 ( 0.005x+ 12t +( pi)/(5))` At x = 1 cm,t =1 sec `y ( 1,1) = 7.5 sin ( 0.005 xx 1 +12 xx 1 + (pi)/(4))` `= 7.5 sin ( 12.005 +( pi)/(4))` ...(1) Now, `theta =( 12.005+ (pi)/(4))` radian `=( 180)/( pi) (12.005+ (pi)/(4)) ` degree `= ( 12.005 xx 180)/( (22)/(7))+ 45 =732.55^(@)` From (1), `y(1,1) = 7.5 sin ( 732.55^(@))` `= 7.5 sin( 720 +12.55^(@))` `= 7.5 sin 12.55^(@) = 7.5 xx 0.2173 =1.63 cm ` velocity of oscillation, `v= ( dy)/(dt)(1,1)` `= (d)/(dt) [ 7.5 sin ( 0.05x+ 12t + ( pi)/(4))]` `= 7.5 xx12 cos ( 0.005x + 12t + (pi)/(4))` At x = 1cm ,t = 1 sec `v = 7.5 xx 12 cos ( 0.005 + 12 + ( pi)/(4))` ` = 90 cos ( 732.55^(@))` `= 90 cos ( 720 + 12.55^(@))` `= 90cos ( 12.55^(@))` `=90 xx 0.9765` `= 87.89 cm//s ` Comparing thegiven equation with the standard form `y ( x,t) = r sin [ ( 2pi)/( lambda) ( vt+x) + phi_(0)]` We get `r= 7.5 `cm`, ( 2piv)/(lambda)=12 ` ( or ) ` 2pi V =12 `. `V =( 6)/( pi ) ` `2 ( pi)/( lambda) = 0.005 ` `:. lambda =( 2pi)/( 0.005) =( 2 xx 3.14 )/( 0.005) = 1256m` Velocity of WAVE propagation, `v = V lambda ` `=( 6)/( pi )xx 12.56 ` `= 24 m //s `. We find thatvelocityat x = 1cm t = 1secis not equal to velocity of wave propagation. (B) Now, all pointswhich are at a distance of `+-lambda, +- 2 lambda, +- 3 lambda ` from x =1 cmwill have same transverse displacement and velocity. As `lambda= 12.56` m, therefore, all POINTS at distances `+- 12.6m , +- 25.2 m` displacement and velocity. As `lambda = 12.56 m`,therefore all points at distance `+-12.6m,+- 25.2m , +- 37m` from x = 11M will havesame displacement & velocityat x= 1cm point at t = 25.55 & 115s. |
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