Question

Longitudi-
nal standing waves can be produced in a solid rod by holding it at
some point between the fingers of one hand and stroking it with the
other hand. The rod oscillates with antinodes at both ends. (a) Why
are the ends antinodes and not nodes? (b) The fundamental fre-
quency can be obtained by stroking the rod while it is held at its
center. Explain why this is the only place to hold the rod to obtain
the fundamental. (c) Calculate the fundamental frequency of a
steel rod of length 1.50 m (see Table 16.1). (d) What is the next
possible standing-wave frequency of this rod? Where should the
rod be held to excite a standing wave of this frequency?

Answer 1

a) Node is formed at the end point of a rod if it is fixed and antinode is formed if the end is free.

So , for the antinode to be produced at the end points of the rod , the end points of the rod must be free.

b) Both the ends of the rod are free. So for the rod to oscillate with the fundamental frequency , a node must be formed at the middle of the rod. So holding the rod at its center makes the node to form at that point.

c) Fundamental frequency of the rod is given by

f = n v / 2 l

where n = 1 is number of loops or modes formed

v = 5941 m/s is the speed of sound wave in steel

l = 1.50 m is the length of the rod

Then , f = 1 * 5941 / ( 2 * 1.50 )

= 1980.33 Hz

d) ( i ) For the next possible frequency n = 2

Then f' = ( 2 * 5941 ) / ( 2 * 1.50 )

= 3960.66 Hz

( ii ) For fundamental frequency ,wavelength = 2l = 3.0 m

For the next possible frequency , = l = 1.50 m

Node and antinode distance = / 4 = 0.375 m

Since the antinode exists at the end point , the rod must be held at a distance of 0.375 m ( point where node exists ) from the edge of the rod