Question 51:
moderateThe two nearest harmonics of a tube closed at one end and open at other end are $220 \text{ Hz}$ and $260 \text{ Hz}$. What is the fundamental frequency of the system?
(2017-Delhi)
For a closed pipe, frequencies are odd multiples of fundamental $f_0$. Difference between successive harmonics is $2f_0 = 260 - 220 = 40 \text{ Hz}$. Thus, $f_0 = 20 \text{ Hz}$.
Question 52:
easyThe second overtone of an open organ pipe has the same frequency as the first overtone of a closed pipe $L$ meter long. The length of the open pipe will be:
(2016-II)
2nd overtone of open pipe = $3 \times (v / 2L_{\text{open}})$. 1st overtone of closed pipe = $3 \times (v / 4L)$. Equating them: $3v / 2L_{\text{open}} = 3v / 4L \Rightarrow L_{\text{open}} = 2L$.
Question 53:
easyIf we study the vibration of a pipe open at both ends, then the following statement is not true:
(2013)
In an open organ pipe, the open ends act as displacement antinodes and pressure nodes. Therefore, the pressure change is minimum (zero) at both ends, making statement 'a' incorrect.
Question 54:
moderateIf the tension and diameter of a sonometer wire of fundamental frequency $n$ is doubled and density is halved then its fundamental frequency will become:
(2001)
Fundamental frequency $n = \frac{1}{lD} \sqrt{\frac{T}{\pi \rho}}$. With new values $T' = 2T$, $D' = 2D$, and $\rho' = \rho/2$, the new frequency $n' = \frac{1}{2D} \sqrt{\frac{2T}{\pi (\rho/2)}} = \frac{1}{2D} \sqrt{\frac{4T}{\pi \rho}} = n$.
Question 55:
easyA string is cut into three parts, having fundamental frequencies $n_1$, $n_2$ and $n_3$ respectively. Then original fundamental frequency ‘$n$’ related by the expression as:
(2000)
For a given string, fundamental frequency $n \propto 1/l$. For the cut segments, $l = l_1 + l_2 + l_3$. Substituting $l = k/n$, we get $\frac{1}{n} = \frac{1}{n_1} + \frac{1}{n_2} + \frac{1}{n_3}$.
Question 56:
easyA standing wave having 3 nodes and 2 antinodes is formed between $1.21 \text{ \AA}$ distance then the wavelength is:
(1998)
A standing wave with 3 nodes and 2 antinodes corresponds to 2 full loops. The length of one loop is $\lambda/2$. Total distance $L = 2 \times (\lambda/2) = \lambda$. Therefore, the wavelength $\lambda = 1.21 \text{ \AA}$.
Question 57:
moderateThe length of a sonometer wire AB is $110 \text{ cm}$. Where should the two bridges be placed from A to divide the wire in 3 segments whose fundamental frequencies are in the ratio of $1:2:3$?
(1995)
Since frequency $f \propto 1/l$, the lengths will be in the ratio $1/1 : 1/2 : 1/3 = 6:3:2$. The sum of the ratios is 11. Lengths are $60 \text{ cm}$, $30 \text{ cm}$, and $20 \text{ cm}$. The bridges should be placed at $60 \text{ cm}$ and $60 + 30 = 90 \text{ cm}$ from A.
Question 58:
moderateA stretched string resonates with tuning fork frequency $512 \text{ Hz}$ when length of the string is $0.5 \text{ m}$. The length of the string required to vibrate resonantly with a tuning fork of frequency $256 \text{ Hz}$ would be:
(1993)
By the law of length for strings, $f_1 L_1 = f_2 L_2$. Substituting the values, $512 \times 0.5 = 256 \times L_2$. Solving for $L_2$ yields $1 \text{ m}$.
Question 59:
moderateA closed organ pipe (closed at one end) is excited to support the third overtone. It is found that air in the pipe has:
(1991)
For a closed pipe, the overtone sequence follows odd harmonics ($1, 3, 5, 7$). The third overtone corresponds to the 7th harmonic. It has 4 nodes and 4 antinodes (number of nodes = number of antinodes = overtone + 1).