Transverse Stationary Waves and Sonometer Questions in English

The length of the string of a musical instrument is $90 \;cm$ and has a fundamental frequency of $120 \;Hz$. Where (in $cm$) should it be pressed to produce a fundamental frequency of $180 \;Hz$?

$A$ wire having a linear mass density $9.0 \times 10^{-4} \; \text{kg/m}$ is stretched between two rigid supports with a tension of $900 \; \text{N}$. The wire resonates at a frequency of $500 \; \text{Hz}$. The next higher frequency at which the same wire resonates is $550 \; \text{Hz}$. The length of the wire is $...... \; \text{m}$.

$A$ wire of length $0.3\,m$,stretched between rigid supports,has its $n^{\text{th}}$ and $(n+1)^{\text{th}}$ harmonics at $400\,Hz$ and $450\,Hz$,respectively. If the tension in the string is $2700\,N$,its linear mass density is ......... $kg/m$.

If the fundamental frequency of a string is $220 \, cps$,the frequency of the fifth harmonic will be ......... $cps$.

The tension in a wire is decreased by $19 \%$. The percentage decrease in frequency will be ......... $\%$

$A$ $12 \,m$ long vibrating string has a wave speed of $48 \,m/s$. At what frequencies will it resonate in $cps$?

$A$ certain string will resonate at several frequencies,the lowest of which is $200 \,Hz$. What are the next three higher frequencies at which it resonates?

The string of a violin has a frequency of $440 \,cps$. If the violin string is shortened by one fifth,its frequency will be changed to ........... $cps$.

$A$ wire of length $1 \,m$ under a certain initial tension emits a sound of fundamental frequency $256 \,Hz$. When the tension is increased by $1 \,kg \,wt$,the frequency of the fundamental note increases to $320 \,Hz$. The initial tension is ........... $kg \,wt$.

$A$ tuning fork vibrating with a sonometer having a wire of length $20 \,cm$ produces $5$ beats per second. The beat frequency does not change if the length of the wire is changed to $21 \,cm$. The frequency of the tuning fork must be ............ $Hz$.

$A$ second harmonic has to be generated in a string of length $l$ stretched between two rigid supports. The points where the string has to be plucked and touched are respectively

$A$ uniform string resonates with a tuning fork at a maximum tension of $32 \,N$. If it is divided into two segments by placing a wedge at a distance one-fourth of the length from one end,then to resonate with the same frequency,the maximum value of tension for the string will be ........... $N$.

The frequency of a sonometer wire is $100 \ Hz$. When the weights producing the tension are completely immersed in water,the frequency becomes $80 \ Hz$,and on immersing the weights in a certain liquid,the frequency becomes $60 \ Hz$. The specific gravity of the liquid is:

$A$ guitar string of length $90\,cm$ vibrates with a fundamental frequency of $120\,Hz.$ The length of the string producing a fundamental frequency of $180\,Hz$ will be $...........cm$.

$A$ wire of density $8 \times 10^3\,kg/m^3$ is stretched between two clamps $0.5\,m$ apart. The extension developed in the wire is $3.2 \times 10^{-4}\,m$. If Young's modulus $Y = 8 \times 10^{10}\,N/m^2$,the fundamental frequency of vibration in the wire will be $......\,Hz$.

In an experiment with a sonometer,when a mass of $180\,g$ is attached to the string,it vibrates with a fundamental frequency of $30\,Hz$. When a mass $m$ is attached,the string vibrates with a fundamental frequency of $50\,Hz$. The value of $m$ is $.........\,g$.

$A$ tuning fork resonates with a sonometer wire of length $1 \ m$ stretched with a tension of $6 \ N$. When the tension in the wire is changed to $54 \ N$,the same tuning fork produces $12$ beats per second with it. The frequency of the tuning fork is $Hz$.

$A$ sonometer wire of resonating length $90 \ cm$ has a fundamental frequency of $400 \ Hz$ when kept under some tension. The resonating length of the wire with a fundamental frequency of $600 \ Hz$ under the same tension is . . . . . . $cm$.

$A$ string of length $1 \,m$ and mass $2 \times 10^{-5} \,kg$ is under tension $T$. When the string vibrates, two successive harmonics are found to occur at frequencies $750 \,Hz$ and $1000 \,Hz$. The value of tension $T$ is . . . . . . Newton.

Answer the following by appropriately matching the lists based on the information given in the paragraph.
$A$ musical instrument is made using four different metal strings,$1, 2, 3$ and $4$ with mass per unit length $\mu, 2\mu, 3\mu$ and $4\mu$ respectively. The instrument is played by vibrating the strings by varying the free length in between the range $L_0$ and $2L_0$. It is found that in string-$1$ $(\mu)$ at free length $L_0$ and tension $T_0$ the fundamental mode frequency is $f_0$.
$List-I$ gives the above four strings while $List-II$ lists the magnitude of some quantity.

$List-I$	$List-II$
$(I)$ String-$1$ $(\mu)$	$(P) 1$
$(II)$ String-$2$ $(2\mu)$	$(Q) 1/2$
$(III)$ String-$3$ $(3\mu)$	$(R) 1/\sqrt{2}$
$(IV)$ String-$4$ $(4\mu)$	$(S) 1/\sqrt{3}$
	$(T) 3/16$
	$(U) 1/16$

$(1)$ If the tension in each string is $T_0$,the correct match for the fundamental frequency in $f_0$ units will be,
$(1)$ $I \rightarrow P, II \rightarrow R, III \rightarrow S, IV \rightarrow Q$
$(2)$ $I \rightarrow P, II \rightarrow Q, III \rightarrow T, IV \rightarrow S$
$(3)$ $I \rightarrow Q, II \rightarrow S, III \rightarrow R, IV \rightarrow P$
$(4)$ $I \rightarrow Q, II \rightarrow P, III \rightarrow R, IV \rightarrow T$
$(2)$ The lengths of the strings $1, 2, 3$ and $4$ are kept fixed at $L_0, 3L_0/2, 5L_0/4$ and $7L_0/4$,respectively. Strings $1, 2, 3$ and $4$ are vibrated at their $1^{st}, 3^{rd}, 5^{th}$ and $14^{th}$ harmonics,respectively,such that all the strings have the same frequency. The correct match for the tension in the four strings in the units of $T_0$ will be.
$(1)$ $I \rightarrow P, II \rightarrow Q, III \rightarrow T, IV \rightarrow U$
$(2)$ $I \rightarrow T, II \rightarrow Q, III \rightarrow R, IV \rightarrow U$
$(3)$ $I \rightarrow P, II \rightarrow Q, III \rightarrow R, IV \rightarrow T$
$(4)$ $I \rightarrow P, II \rightarrow R, III \rightarrow T, IV \rightarrow U$

$A$ horizontal stretched string,fixed at two ends,is vibrating in its fifth harmonic according to the equation,$y(x, t) = (0.01 \ m) \sin[(62.8 \ m^{-1}) x] \cos[(628 \ s^{-1}) t]$. Assuming $\pi = 3.14$,the correct statement$(s)$ is (are) :
$(A)$ The number of nodes is $5$.
$(B)$ The length of the string is $0.25 \ m$.
$(C)$ The maximum displacement of the midpoint of the string from its equilibrium position is $0.01 \ m$.
$(D)$ The fundamental frequency is $100 \ Hz$.

$A$ wire stretched between two rigid supports vibrates in its fundamental mode with a frequency of $25 \ Hz$. The mass of the wire is $2 \ g$ and its linear mass density is $4 \times 10^{-3} \ kg/m$. What is the tension in the string (in $N$)?

Two vibrating strings $A$ and $B$ of the same material but lengths $3 L$ and $2 L$ have radii $2 r$ and $3 r$ respectively. They are stretched under the same tension. String $A$ vibrates in the second overtone and string $B$ in the fundamental mode. The ratio of their frequencies $n_A / n_B$ will be:

Length of a sonometer wire is either $95 \ cm$ or $100 \ cm$. In both the cases,a tuning fork produces $5 \ beats/sec$ with the wire. The frequency of the tuning fork is- (in $Hz$)

$A$ tuning fork produces $4$ beats per second when sounded with a sonometer wire of vibrating length $48 \ cm$. It also produces $4$ beats per second when the vibrating length is $50 \ cm$,keeping the tension in the wire the same. What is the frequency of the tuning fork in $Hz$?

$A$ steel wire of length $1 \ m$ and mass $0.1 \ kg$ and having a uniform cross-sectional area of $10^{-6} \ m^2$ is rigidly fixed at both ends. The temperature of the wire is lowered by $20^{\circ} C$. If the wire is vibrating in its fundamental mode,find the frequency (in $Hz$).
$(Y_{\text{steel}} = 2 \times 10^{11} \ N/m^2, \alpha_{\text{steel}} = 1.21 \times 10^{-5} /^{\circ} C)$

$A$ violin emits sound waves of frequency $n_1$ under tension $T$. When tension is increased by $44\%$,keeping the length and mass per unit length constant,the frequency of sound waves becomes $n_2$. The ratio of frequency $n_2$ to frequency $n_1$ is:

When a string of length $l$ is divided into three segments of length $l_1, l_2$ and $l_3$,the fundamental frequencies of the three segments are $n_1, n_2$ and $n_3$ respectively. The original fundamental frequency $n$ of the string is:

$A$ stretched wire of length $260 \ cm$ is set into vibrations. It is divided into three segments whose frequencies are in the ratio $2:3:4$. Their lengths must be

$A$ sonometer wire $49 \ cm$ long is in unison with a tuning fork of frequency '$n$'. If the length of the wire is decreased by $1 \ cm$ and it is vibrated with the same tuning fork,$6$ beats are heard per second. The value of '$n$' is (in $Hz$)

At the poles of the earth,a stretched wire of a given length vibrates in unison with a tuning fork. At the equator of the earth,for the same setting,to produce resonance with the same fork,the vibrating length of the wire

At the poles,a stretched wire of a given length vibrates in unison with a tuning fork. At the equator,for the same setting to produce resonance with the same fork,the vibrating length of the wire

In a sonometer experiment,a string of length $L$ under tension vibrates in its second overtone between two bridges. The amplitude of vibration is maximum at:

$A$ stationary wave is produced along a stretched string of length $80 \,cm$. The resonant frequencies of the string are $90 \,Hz$, $150 \,Hz$, and $210 \,Hz$. The speed of the transverse wave in the string is: (in $\,m/s$)

In Melde's experiment,when the tension decreases by $0.009 \ kg-wt$,the number of loops changes from $4$ to $5$. The initial tension is

If you set up the $7^{th}$ overtone on a string fixed at both ends,how many nodes and antinodes are set up in it?

$A$ uniform wire of length $L$,diameter $D$,and density $\rho$ is stretched by a tension $T$. The frequency $f$ of the wire is proportional to:

$A$ wire of length $L$,diameter $d$,and density of material $\rho$ is under tension $T$,having a fundamental frequency of vibration $n_A$. Another wire of length $2L$,tension $2T$,density $2\rho$,and diameter $3d$ has a fundamental frequency of vibration $n_B$. The ratio $n_B : n_A$ is

Two uniform wires of same material are vibrating under the same tension. If the $1^{\text{st}}$ overtone of the $1^{\text{st}}$ wire is equal to the $2^{\text{nd}}$ overtone of the $2^{\text{nd}}$ wire and the radius of the $1^{\text{st}}$ wire is twice the radius of the $2^{\text{nd}}$ wire,find the ratio of the length of the $1^{\text{st}}$ wire to that of the $2^{\text{nd}}$ wire.

The fundamental frequency of a sonometer wire is $50 \text{ Hz}$ for a given length and tension. If the length is increased by $25 \%$ while keeping the tension constant,what is the percentage change in the frequency of the second harmonic?

Two tuning forks when sounded together produce $4$ beats per second. One of the forks is in unison with $23 \ cm$ length of a sonometer wire and the other with $24 \ cm$ length of the same wire. The frequencies of the two tuning forks are

The fundamental frequency of a sonometer wire is $n$. If the tension is increased $3$ times,the length is increased $3$ times,and the diameter is increased $2$ times,what will be the new frequency?

At poles,a stretched wire of a given length vibrates in unison with a tuning fork. At the equator,for the same setting,to produce resonance with the same fork,the vibrating length of the wire

$A$ tuning fork gives $5$ beats per second with $40 \ cm$ length of a sonometer wire. If the length of the wire is shortened by $1 \ cm$,the number of beats is still the same. The frequency of the fork is (in $Hz$)

In a Sonometer experiment,the frequency of the tuning fork used is $288 \ Hz$. Harmonics will '$NOT$' be produced at the frequency: (in $Hz$)

The fundamental frequency of a sonometer wire is $n$. If the tension is increased $3$ times,the length is increased $3$ times,and the diameter is increased $2$ times,what will be the new frequency?

$A$ string $A$ has twice the length,twice the diameter,twice the tension,and twice the density of another string $B$. The overtone of $A$ which will have the same fundamental frequency as that of $B$ is:

$A$ sonometer wire is stretched by hanging a metal bob,the fundamental frequency of the wire is $n_1$. When the bob is completely immersed in water,the frequency of vibration of the wire becomes $n_2$. The relative density of the metal of the bob is

When the tension in a string is increased by $3 \ kg \ wt$,the frequency of the fundamental mode increases in the ratio $2:3$. The initial tension in the string is: (in $kg \ wt$)

$A$ sonometer wire is in unison with a tuning fork of frequency '$n$' when it is stretched by a weight of specific gravity '$d$'. When the weight is completely immersed in water,'$x$' beats are produced per second,then