Question

A string \[2.0m\] long and fixed at its ends is driven by a 240 Hz vibrator. The string vibrates in its third harmonic mode. The speed of the wave and its fundamental frequency are?
(A) 320 m/s, 120 Hz
(B) 180 m/s, 80 Hz
(C) 180 m/s, 120 Hz
(D) 320 m/s, 80 Hz

Answer 1

Hint: A string fixed at its ends signifies that it has nodes at its ends. The third harmonic mode has a frequency which is simply thrice the frequency of the fundamental mode.
Formula used: In this solution we will be using the following formulae;
\[{f_0} = \dfrac{v}{{2l}}\] where \[{f_0}\] is the fundamental frequency of string fixed at both ends. \[v\] is the speed of the wave on the string, and \[l\] is the length of the string.
\[{f_2} = 3{f_0}\] where \[{f_2}\] is the third harmonic frequency (also known as the second overtone frequency)

Complete Step-by-Step Solution:
A particular vibrator at a particular frequency is said to be vibrating a string of a particular length and fixed at its ends. This string is said to vibrate at its third harmonic mode (also commonly called second overtone mode). We are to determine the speed of the wave and the fundamental frequency.
To calculate the fundamental frequency, we shall note that the third harmonics can be given as
\[{f_2} = 3{f_0}\] where \[{f_2}\] is the third harmonic frequency (also known as the second overtone frequency) and \[{f_0}\] is the fundamental frequency, hence,
\[{f_0} = \dfrac{{{f_2}}}{3}\] which by inserting given values, we get,
\[{f_0} = \dfrac{{240}}{3} = 80Hz\]
Now to calculate the velocity, we recall that the fundamental frequency can be given as
\[{f_0} = \dfrac{v}{{2l}}\] where \[v\] is the speed of the wave on the string, and \[l\] is the length of the string.
Hence, we get
\[v = 2l{f_0} = 2 \times 2 \times 80\]
\[ \Rightarrow v = 320m/s\]

Hence, the correct option is D

Note: Alternatively, we could calculate first, the speed of light from
\[{f_2} = 3{f_0} = 3\dfrac{v}{{2l}}\]
Hence, \[v = \dfrac{{2l{f_2}}}{3}\] which by inserting values, we get,
\[v = \dfrac{{2 \times 2 \times 240}}{3} = 320m/s\]
And then the fundamental frequency can be calculated as
\[{f_2} = \dfrac{v}{{2l}} = \dfrac{{320}}{{2 \times 2}} = 80Hz\]