Question

The wave function for a certain standing wave on a string fixed at both ends is y(x, t) = 0.5$\sin \left(0.025\pi x\right)\cos 500t$. Where x and y are in centimetres and t is in seconds. The shortest possible length of the string is

• A. 126 cm
• B. 160 cm
• C. 40 cm
• D. 80 cm

Answer 1

The correct option is C 40 cm

For the shortest possible length, it should be allowing fundamental frequency resonance.

In fundamental frequency,

$L=\frac{\lambda }{2}L=\frac{2\pi /k}{2}(\because k=\frac{2\pi }{\lambda })L=\frac{\pi }{k}$

From $y=0.5\sin \left(0.025\pi x\right)\cos \left(500t\right)k=0.025\pi$

(on comparing with $y=A\sin \left(kx\right)\cos \left(\omega t\right)$)

$\therefore L=\frac{\pi }{0.025\pi }=\frac{1000}{25}[L=40cm]$

Hence option (C) is correct