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As found in case of string , in case of rods also clamped point behaves as a
node while the free end antinode . the situation is shown in figure i have
made for you in paint : -)
Because the distance between two consecutive nodes is ( lambda/2)
while between a node and antinode is lambda/4hence
4× { lamda/2} + 2 {lambda/4} = > L or lambda = > 2×1/5 = > 0.4m
further , it is given that
lambda = > 2×10^11 N/m² andρ = > 8×10³kg/m³
v= >√(λ/ρ) = >√[(2×10^11)/8×10^3] = > 5000 m /s
Hence fromν = > nλ
n = > v/λ = > 5000/0.4 = > 125 Hz
Now if incident and reflected waves along the rod are
y1 = > A sin(ωt -kx) and y2 = >sin(ωt+kx+Φ)
THE RESULTANT WAVE WILL BE
y = > y1 +y2 => Asin (ωt-kx) + sin(ωt+kx+Φ) = > 2A cos ( kx + (Φ/2) )sin (ωt+(Φ/2)
Dear Student,
Please find below the solution to your problem.
Because there is an antinode at the free end of the rod , hence amplitude is maximum at x = > 0 so
cos ( k×0 + (Φ/2) ) = > maximum = > 1 i.e Φ = > 0
And ,
A max = > 2A = > 2×10^-6 m ------------------------( given )
y = > 2×10^-6 cos kx sinωt
as we know that y => 2×10^-6 cos 5πx sin 25000πt
Now because for a point 2cm from the midpoint x = > (0.50+,-0.02) ,
and hence we got
y => 2×10^-6 Cos 5π(0.5 +,- 0.02) sin25000(πt) is our required answer
Thanks and Regards
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