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Asked by sarveshvibrantacademy | 02 Apr, 2019, 14:43: PM
answered-by-expert Expert Answer
Displacement Y at x and at a time t is given by  Y =  2a sin(kx) cos(ωt) ..............................(1)
 
Given equation for displacement  Y = 2 sin(πx) sin( 100πt ) = 2 sin(πx) cos[100πt-(π/2)] ......................(2)
 
(A) by comparing (1) and (2), we have maximum displacement , 2a sin(kx) = 2 sin(πx)
 
maximum displacement at x= 1/6 cm ,  2sin(π/6) = 1 mm

 
(B) velocity of the particle is obtained by differentiating eqn.(2),
 
dY/dt = -2 sin(πx) 100π sin[100πt-(π/2)]
 
velocity at x = π/6  cm and at t= 1/600 s  is  = -2 (1/2) 100π sin[ (π/6) - (π/2) ]  = 100π (√3/2) = 157√3 mm/s
 
 
(C) By comparing eqn.(1) and eqn.(2), we get k = 2π/λ = π  or  λ = 2 cm
 
if L is length , for standing wave we have  L = nλ/2  , where n = 1,2,3.....
 
if L = 10 cm and λ = 2 cm , we get n = number of loops = 10
Answered by Thiyagarajan K | 02 Apr, 2019, 15:33: PM

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