String Fixed at Both Ends

Q.

A string is stretched between fixed points separated by 75 cm. It is observed to have resonant frequencies of 420 Hz and 315 Hz. There are no other resonant frequencies between these two. Then, the lowest resonant frequency for this string

Q. Standing waves are produced in a 10 m long stretched string. If the string vibrates in 5 segments and the wave velocity is 20 m/s, the frequency is

1. 5 Hz
2. 10 Hz
3. 2 Hz
4. 4 Hz

Q. The fundamental frequency of a sonometer wire increases by $6\text{Hz}$ if its tension is increased by $44\%$, keeping the length constant. What is the change in the fundamental frequency of the sonometer wire when the length of the wire 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 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 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. If $n_1,n_2$ and $n_3$ are the fundamental frequencies of three segments of a string of length $l$, then the original fundamental frequency $n$ of the string is given by

1. $\frac{1}{n}=\frac{1}{n_1}+\frac{1}{n_2}+\frac{1}{n_3}$
2. $\sqrt{n}=\sqrt{n_1}+\sqrt{n_2}+\sqrt{n_3}$
3. $n=n_1+n_2+n_3$
4. $\frac{1}{\sqrt{n}}=\frac{1}{\sqrt{n_1}}+\frac{1}{\sqrt{n_2}}+\frac{1}{\sqrt{n_3}}$

Q. A massless rod $BD$ is suspended by two identical massless strings $AB$ and $CD$ of equal lengths. A block of mass $m$ is suspended from point $P$ such that $BP$ is equal to $x$. If the fundamental frequency of the left wire is twice the fundamental frequency of right wire, then the value of $x$ is

1. $\frac{3l}{4}$
2. $\frac{4l}{5}$
3. $\frac{l}{5}$
4. $\frac{l}{4}$

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

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

Q.

What are normal modes of oscillation?

Q. A string is hanging from a rigid support. A transverse wave pulse is setup at the free end. The velocity $v$ of the pulse related to the distance $x$ covered by it is

1. $v\propto \sqrt{x}$
2. $v\propto x$
3. $v\propto \frac{1}{x}$
4. $v\propto \frac{1}{x^2}$

Q. An air column in a pipe closed at one end is made to vibrate in its second overtone by a tuning fork of frequency $440Hz$. The speed of sound wave in air is $330m/s$. End corrections may be neglected. Let $P_0$ denote the mean pressure at any point in the pipe and $\Delta P_0$ be the maximum amplitude of pressure variation. Then

1. length of the pipe is $\frac{15}{16}m$
2. length of the pipe is $\frac{9}{16}m$
3. the maximum pressure at the open end is $P_0$
4. the minimum pressure at the open end is $P_0$

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. When the string of a sonometer of length $L$ between the bridges vibrates in the second overtone, the amplitude of vibration is maximum at

1. $\left(\frac{L}{2}\right)$
2. $\left(\frac{L}{4}\right)$ and $\left(\frac{3L}{4}\right)$
3. $\frac{L}{6},\frac{L}{2}$ and $\frac{5L}{6}$
4. $\frac{L}{8},\frac{3L}{8},\frac{5L}{8}$

Q. When a wave travels in a medium, the particle displacement is given by $y=a\sin 2\pi (bt-cx),$ where $a,b$ and $c$ are constants. The maximum particle velocity will be twice the wave velocity if

1. $c=\frac{1}{\pi a}$
2. $c=\pi a$
3. $b=ac$
4. $b=\frac{1}{ac}$

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 $4.0\times {10}^{-2}{\text{kg m}}^{-1}$. What is the tension in the string?

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

Q.

A sono-meter wire supports a 4 kg load and vibrates in fundamental node with a tuning fork of frequency 416 Hz. The length of the wire between the bridges is now doubled. In order to maintain fundamental node, the load should be changed to

1. 1 kg

2. 16 kg

3. 2 kg

4. 8 kg

Q.

Statement I: A speech signal of $2kHz$ is used to modulate a carrier signal of $1MHz$. The bandwidth requirement for the signal is $4kHz$.

Statement II: The sideband frequencies are $1002kHz$ and $998kHz$. In the light of the above statements, choose the correct answer from the options given below:

1. Both statement I and statement II are false

2. Statement I is false but statement II is true

3. Statement I is true but statement II is false

4. Both statement I and statement II are true

Q. The length of the wire between the pulleys is $1.5\text{m}$ and its mass is $12\text{gm}$. Find the frequency of vibration, with which the wire vibrates in two loops leaving the middle point of the wire between the pulleys at rest.
Use value of $\sqrt{11250}\approx 106$

1. $70.6\text{Hz}$
2. $80.7\text{Hz}$
3. $35.6\text{Hz}$
4. $40.3\text{Hz}$

Q.

A 1cm long string vibrates with fundamental frequency of 256 Hz. If the length is reduced to $\frac{1}{4}cm$ keeping the tension unaltered, the new fundamental frequency will be (in Hz) (string is fixed at both ends)

1. 64
2. 1024
3. 256
4. 512

Q. A wire (fixed at both ends) of length $l$ having tension $T$ and radius $r$ vibrates with natural frequency $f$. Another wire of same metal with length $2l$ having tension $2T$ and radius $2r$ will vibrate with natural frequency

1. $2f$
2. $2\sqrt{2}f$
3. $\frac{f}{2\sqrt{2}}$
4. $f$

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

Q. A string fixed at both ends is vibrating in the lowest mode of vibration for which a point at quarter of its length from one end is a point of maximum displacement. The frequency of vibration in this mode is 100 Hz. The frequency emitted when it vibrates in the next mode, such that this point is again a point of maximum displacement is $\eta \times 100$ Hz, where the value of $\eta$ is

Q. In the experimental setup of metre bridge shown in the figure, the null point is obtained at a distance of $40\text{cm}$ from $\text{A}$. If a $10\Omega$ resistor is connected in series with $R_1\Omega$ , the null point is observed to shift by $10\text{cm}$ from $\text{A}$. Find the resistance that should be connected in parallel with $(R_1+10)$$\Omega$ such that the null point shifts back to its initial position is

1. $120\Omega$
2. $90\Omega$
3. $60\Omega$
4. $30\Omega$

Q.

In a resonance tube, using a tuning fork of frequency 325 Hz, two successive resonance lengths are observed as 25.4 cm and 77.4 cm respectively. The velocity of sound in air

1. 338 ms^-1

2. 328 ms^-1

3. 330 ms^-1

4. 320 ms^-1

Q. The frequency of a sonometer wire is $100\text{Hz}$. When the weight producing the tension is completely immersed in water the frequency becomes $80\text{Hz}$ and on immersing the weight in a certain liquid the frequency becomes $60\text{Hz}$. The specific gravity of the liquid is

1. $1.42$
2. $1.77$
3. $1.82$
4. $1.21$

Q.

A wire of length $L$ and mass $6\times {10}^{-3}kg{m}^{-1}$ per unit length is put under the tension of $540N$. Two consecutive frequencies that it resonates at are: $420Hz$ and $490Hz$ . Then $L$ in meter is

1. $8.1m$

2. $2.1m$

3. $1.1m$

4. $5.1m$

Q.

How is the frequency of a stretched string related to its length?

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