String Fixed at Both Ends

Q.

What happens to frequency when the mass is doubled?

Q.

A cylindrical tube open at both ends has a fundamental frequency$f$ in air. The tube is dipped vertically in water so that half of it is in water. The fundamental frequency of the air column is now

1. $\frac{f}{2}$

2. $f$

3. $\frac{3f}{4}$

4. $2f$

Q.

A pipe closed at one end produces a fundamental note of $412Hz$. It is cut into two pieces of equal length. The fundamental frequencies produced in the two pieces are

1. $206Hz,412Hz$

2. $824Hz,1648Hz$

3. $412Hz,824Hz$

4. $206Hz,824Hz$

Q. The fundamental frequency of a sonometer wire increases by $6\text{Hz}$ if the tension in the wire is increased by $44\%$, keeping the length constant. What is the change in the fundamental frequency of the wire when the length is increased by $20\%$, keeping the original tension in the wire constant.

1. $5\text{Hz}$
2. $10\text{Hz}$
3. $2.5\text{Hz}$
4. $7.5\text{Hz}$

Q. Two strings A and B made of same material, have lengths $l_A$ and $l_B$ and are connected to two other comparitively denser medium strings having masses $M_A$ and $M_B$, the upper ends being supported by rigid supports. If $f_A$ and $f_B$ are their fundamental frequencies of their vibrations, then $f_A:f_B$ is

1. $\frac{M_A{l}_A}{M_B{l}_B}$
2. $\frac{l_A}{l_B}\sqrt{\frac{M_A}{M_B}}$
3. $\frac{M_A}{M_B}\sqrt{\frac{l_A}{l_B}}$
4. None of these

Q. A standing wave is created on a string fixed at both ends, of length $120\text{cm}$ and it is vibrating in $6^{th}$ harmonic. Maximum possible amplitude of any particle is $10\text{cm}$ and maximum possible velocity will be $10\text{cm/s}$. Which of the following statement(s) is/are correct?

1. Angular wave number $\left(k\right)$ of the wave will be $\frac{\pi }{20}{\text{cm}}^{-1}$.
2. Angular wave number $\left(k\right)$ of the wave will be $\frac{\pi }{10}{\text{cm}}^{-1}$.
3. Time period of any particles SHM will be $4\pi \text{sec}$.
4. Time period of any particles SHM will be $2\pi \text{sec}$.

Q. A sonometer wire resonates with a given tuning fork, forming standing waves with five antinodes between the two bridges when a mass of $9\text{kg}$ is suspended from the wire. When a mass of $M\text{kg}$ is suspended from the wire, the same tuning fork causes the wire to form three antinodes for the same positions of the bridges. What is the value of $M$?

1. $25\text{kg}$
2. $5\text{kg}$
3. $12.5\text{kg}$
4. $\frac{1}{25}\text{kg}$

Q. A wire of length $2L,$ is made by joining two wires $\text{A}$ and $\text{B}$ made of the same material having same length but different radii $r$ and $2r$. It is vibrating at a frequency such that the joint of the two wires forms a node. If the number of antinodes in wire $\text{A}$ is $p$ and that in $\text{B}$ is $q$, then the ratio $p:q$ is:

1. $3:5$
2. $1:2$
3. $4:9$
4. $1:4$

Q. Assertion: If a stretched wire fixed at both ends is vibrating in its second overtone mode, then total number of nodes and antinodes are four each.
Reason: If a stretched wire fixed at both ends is vibrating, number of antinodes are equal to number of loops.
You are required to choose correct one of the following

1. Both Assertion and Reason are correct and Reason is the correct explanation of Assertion.
2. Both Assertion and Reason are true but Reason is not the correct explanation of Assertion.
3. Assertion is true but Reason is false.
4. Assertion is false but Reason is true.

Q.

A piano wire weighing $6g$and having a length of $90cm$ emits a fundamental frequency corresponding to the “Middle C” $(\nu =261.63Hz)$. Find the tension in the wire.

Q. The vibrations of a string of length $20\text{cm}$ fixed at both ends are represented by the equation $y=2\sin \left(\frac{\pi x}{5}\right)\cos \left(12\pi t\right)$, where $x$ and $y$ are in $\text{cm}$ and $t$ in $\text{sec}$. How many nodes are located along the string?

1. $3$
2. $4$
3. $5$
4. $6$

Q. In a stationary wave

1. strain is maximum at nodes.
2. strain is minimum at nodes.
3. strain is maximum at antinodes.
4. amplitude is zero at all points.

Q. A wire of density $9\times {10}^{-3}{\text{kg cm}}^{-3}$ is stretched between two clamps $1\text{m}$ apart. The resulting strain in the wire is $4.9\times {10}^{-4}$. The lowest frequency of the transverse vibrations in the wire (in $\text{Hz}$) is

(Young's modulus of wire $Y=9\times {10}^{10}{\text{Nm}}^{-2})$

Q. A wire having a linear mass density $5\times {10}^{-3}\text{kg/m}$ is stretched between two rigid supports with a tension of $450\text{N}$. The wire resonates at a frequency of $420\text{Hz}$. The next higher frequency at which the same wire resonates is $490\text{Hz}$. The length of the wire will be

1. $4\text{m}$
2. $2\text{m}$
3. $8\text{m}$
4. $3\text{m}$

Q. The length of a sonometer wire is $1.21\text{m}$. It is divided into three segments. What should be the lengths of the three segments for their fundamental frequencies to be in the ratio $1:2:3$?

1. $0.66\text{m},0.33\text{m},0.22\text{m}$
2. $0.72\text{m},0.30\text{m},0.18\text{m}$
3. $0.56\text{m},0.44\text{m},0.21\text{m}$
4. $0.82\text{m},0.18\text{m},0.21\text{m}$

Q. A string fixed at both ends has consecutive standing wave modes for which the distances between adjacent nodes are $18\text{cm}$ and $16\text{cm}$ respectively. The minimum possible length of the string is:

1. $144\text{cm}$
2. $152\text{cm}$
3. $176\text{cm}$
4. $200\text{cm}$

Q. The frequency of a sonometer wire is $300\text{Hz}$. The frequency becomes half when the mass producing the tension in the wire is completely immersed in water and on immersing the mass in a different liquid, the frequency become one-third. The relative density of the liquid is (upto 2 decimals )

Q. Two vibrating strings of same material but lengths $L$ and $2L$ have radii $2r$ and $r$ respectively. They are stretched under the same tension and vibrate in their fundamental modes with frequencies $f_1$ and $f_2$ respectively. The ratio of $\frac{f_1}{f_2}$ is given by (strings are fixed at both ends)

1. 2
2. 4
3. 8
4. 1

Q. In order to double the frequency of the fundamental note emitted by a stretched string, the length is reduced to ${\left(\frac{3}{4}\right)}^{\text{th}}$ of the original length (mass per unit length remains same) and the tension is changed. The factor by which the tension is to be increased is

1. $\frac{3}{8}$
2. $\frac{2}{3}$
3. $\frac{8}{9}$
4. $\frac{9}{4}$

Q.

A tuning fork of frequency 480 Hz is used to vibrate a sono-meter wire having natural frequency 240 Hz. The wire will vibrate with a frequency of

1. 240 Hz

2. 480 Hz

3. 720 Hz

4. will not vibrate

Q.

Three resonant frequencies of a string are 90, 150 and 210 Hz.

(i) Find the highest possible fundamental frequency of vibration of this string

(ii) Which harmonics of the fundamental are the given frequencies?

(iii) Which overtones are these frequencies?

1. 30Hz, harmonics - 3rd, 5th, 7th, overtones - 2nd, 4th, 6th

2. 60 Hz, harmonics - 3rd, 5th, 7th, overtones - 2nd, 4th, 6th

3. 30 Hz, harmonics - 2nd, 4th, 6th, overtones, 3rd, 5th, 7th

4. None of these

Q. A sonometer wire of length l vibrates in fundamental mode when excited by a tuning fork of frequency 416 Hz. If the length is doubled keeping other things same, the string will

1. vibrate with a frequency of 416 Hz
2. vibrate with a frequency of 208 Hz
3. vibrate with a frequency of 832 Hz
4. stop vibrating.

Q. The length of the wire shown in figure between the pulley and fixed support is $1.5\text{m}$ and mass is $12.0\text{g}$. The frequency of vibration with which the wire vibrates in two loops leaving the middle point of the wire between the pulley and the fixed support, is: [Assume the pulley acts as a fixed support and consider $g=10{\text{m/s}}^2$]

1. $10\text{Hz}$
2. $30\text{Hz}$
3. $100\text{Hz}$
4. $70\text{Hz}$

Q. A steel wire of length $116\text{cm}$ is connected to an aluminum wire of length $150\text{cm}$ and stretched between two fixed supports. The tension produced is $52\text{N}$. The cross-section of each wire is $2{\text{mm}}^2$. If a transverse wave is set up in the wire, find the lowest frequency for which standing waves with a node at the joint are produced. (density of aluminum $\left({\rho }_{al}\right)=2.6{\text{g/cm}}^3$ and density of steel $\left({\rho }_s\right)=7.8{\text{g/cm}}^3)$.

1. $50\text{Hz}$
2. $100\text{Hz}$
3. $150\text{Hz}$
4. $200\text{Hz}$

Q. A string of length $2\text{m}$ is fixed at both ends. If the speed of transverse wave on the string be $500\text{m/s}$, then what will be the frequency of vibration of string in its fifth harmonic and corresponding wavelength of the wave?

1. $125\text{Hz},0.5\text{m}$
2. $625\text{Hz},0.8\text{m}$
3. $500\text{Hz},1.25\text{m}$
4. $250\text{Hz},0.5\text{m}$

Q. In order to double the frequency of the fundamental note emitted by a stretched string, the length is reduced to ${\left(\frac{3}{4}\right)}^{\text{th}}$ of the original length (mass per unit length remains same) and the tension is changed. The factor by which the tension is to be increased is

1. $\frac{3}{8}$
2. $\frac{2}{3}$
3. $\frac{8}{9}$
4. $\frac{9}{4}$

Q. A string of length $L$ vibrates with a frequency of $7.5\text{Hz}$ to form a standing wave with three antinodes as shown below.


What is the oscillating frequency of this string, if it produces the standing wave pattern shown below?

1. $10\text{Hz}$
2. $7.5\text{Hz}$
3. $2.5\text{Hz}$
4. $15\text{Hz}$

Q. A wire stretched between two rigid supports vibrates in its fundamental mode with a frequency of $45\text{Hz}$. The mass of the wire is $3.5\times {10}^{-2}\text{kg}$ and its linear density is $4.0\times {10}^{-2}{\text{kg m}}^{-1}$. The tension in the string is approximately -

1. $300.47\text{N}$
2. $425.25\text{N}$
3. $247.27\text{N}$
4. $500\text{N}$

Q.

The fundamental frequency of a segment of wire vibrating is $450Hz$ and it is under a tension of $9kg–wt$. Then, the tension at which the fundamental frequency of the same wire becomes$900Hz$ is

1. $36kg-wt$

2. $27kg-wt$

3. $18kg-wt$

4. $72kg-wt$

Q. The fundamental frequency of a sonometer wire increases by 6 Hz, if its tension is increased by 44%, keeping the length constant. Find the change in the fundamental frequency of the sonometer wire when the length of the wire is increased by 40%, keeping the original tension in the wire constant.

1. 8.6 Hz
2. 21.4 Hz
3. 51.4 Hz
4. 17.2 Hz