Question

A standing wave in a string 35 cm long has a total of six nodes including those at the ends. Hence wavelength of the standing wave is:
A. 5.8 cm
B. 4.6 cm
C. 10.4 cm
D. 14 cm

Answer 1

Hint:
Since, there are a total of six nodes, we can say that there are five segments between the nodes. The distance between two nodes will be half the wavelength for standing waves on the string. The formula for calculating length of a string is used here.
Formula used:
Length of a string is given by,
     \[L=\left( n-1 \right)\times \lambda /2\]
Where n is the number of nodes and \[\lambda \]is wavelength.

Complete answer:
Given,
     L = 35 cm
     n = 6
Substituting these values in the formula for length, we get,
     \[\begin{gathered}
  & L=\left( n-1 \right)\times \lambda /2 \\
 & 35=\left( 6-1 \right)\times \lambda /2 \\
\end{gathered}\]
Now, we simplify this to find the value of \[\lambda \] as follows:
     \[\begin{gathered}
  & 35=\left( 6-1 \right)\times \lambda /2 \\
 & 35=5\times \lambda /2 \\
 & \lambda =\dfrac{35\times 2}{5} \\
 & \lambda =14 \\
\end{gathered}\]
Therefore, the wavelength of the standing wave is 14 cm.

The answer is option D.

Note:
We should not get confused between the number of nodes and the segments between the nodes. Wavelength is calculated using the sections between the nodes. For example, a string on an instrument is clamped at both ends, and therefore, a string must have a node at each end when it vibrates. However, for a fundamental harmonic, length is calculated as \[\lambda /2\]. This is possible only if we consider the section between the two nodes and not the number of nodes. The wavelength will have the same unit as the given length, that is cm.