Question

The vibrations of a string fixed at both ends are described by the

$y=\left(5.00mm\right)sin\left[\right(1.57c{m}^{-1}\left)x\right]sin\left[\right(314s^{-1}\left)t\right]$

If the length of the string is 10.0 cm, locate the nodes and the antinodes. How many loops are formed in the vibration?

• A. nodes - 0, 2cm, 4 cm, 6 cm, 8 cm, 10 cm Anti-nodes - 1 cm, 3 cm, 5 cm, 7 cm, 9 cm
• B. nodes - 0, 1 cm, 3 cm, 9 cm Anti-nodes - 2 cm, 4 cm, 6 cm, 8 cm, 10 cm
• C. nodes - 0, 4 cm, 8 cm, 10 cm Anti-nodes - 1 cm, 5 cm, 9 cm
• D. None of these

Answer 1

The correct option is A

nodes - 0, 2cm, 4 cm, 6 cm, 8 cm, 10 cm

Anti-nodes - 1 cm, 3 cm, 5 cm, 7 cm, 9 cm

The nodes occur where the amplitude is zero, i.e.,

$sin\left(1.57c{m}^{-1}\right)x=0$

or $\left(\frac{\pi }{2}c{m}^{-1}\right)x=n\pi$

where n is an integer.

Thus,$x=2ncm$

The nodes, therefore, occur at$x=0,2cm,4cm,6cm,8cmand10cm$. antinodes occur in between them, i.e., at $x=1cm,3cm,5cm,7cmand9cm$. The string vibrates in 5 loops.