Q.A steel rod $100\ \text{cm}$ long is clamped at its middle. The fundamental frequency of longitudinal vibrations of the rod are given to be $2.53\ \text{kHz}$. What is the speed of sound in steel?

Showing the 12 most recent of 22 on this concept.

1. TG EAPCET 2026Set eng-2026-05-09-AN1 markMCQ
   Q.The length of an open pipe is half of the length of another closed pipe. When the two pipes are vibrated, four nodes are formed in both the cases. The ratio of the frequencies of the open and the closed pipes is (A) $1 : 1$ (B) $16 : 7$ (C) $16 : 3$ (D) $7 : 3$

   ›Reveal solutionSolution

   The key idea is to relate the mode number (number of nodes) to the harmonic for each pipe type, then use the given length relation to find the frequency ratio. The ratio is $16:7$.

   The problem gives two pipes — one open at both ends, one closed at one end — and tells you that when each is vibrating, exactly four nodes are formed. It also says the length of the open pipe is half the length of the closed pipe. You need the ratio of their frequencies.

   The first step is always to connect the number of nodes to the harmonic (mode number) for each type of pipe. This is where students often slip up, because the node patterns are different for open and closed pipes.

   1. Nodes in an open pipe An open pipe has antinodes at both ends. For the fundamental mode (first harmonic), there is one node in the middle — that’s 1 node. For the second harmonic, there are 2 nodes. In general, for the $n$-th harmonic of an open pipe, the number of nodes is $n$. Here, four nodes are formed, so $n = 4$. The frequency of the $n$-th harmonic of an open pipe of length $L_o$ is

   $f_o = \frac{n v}{2 L_o} = \frac{4 v}{2 L_o} = \frac{2v}{L_o}.$

   2. Nodes in a closed pipe A closed pipe has a node at the closed end and an antinode at the open end. The fundamental (first harmonic) has 1 node (at the closed end). The next possible mode (third harmonic) has 2 nodes. In general, for a closed pipe, only odd harmonics exist: the $m$-th harmonic (where $m = 1,3,5,\dots$) has $\frac{m+1}{2}$ nodes. We are told four nodes are formed. So

   $\frac{m+1}{2} = 4 \quad\Rightarrow\quad m = 7.$

   Thus the pipe is vibrating in its 7th harmonic.

   The frequency of the $m$-th harmonic of a closed pipe of length $L_c$ is …

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2. TG EAPCET 2026Set eng-2026-05-10-AN1 markMCQ
   Q.If a 200 cm long steel rod clamped at its middle is vibrated in its fundamental mode with a frequency of 1.25 kHz, then the speed of longitudinal waves in the rod is (A) $1.25\ \mathrm{km\,s^{-1}}$ (B) $2.5\ \mathrm{km\,s^{-1}}$ (C) $5\ \mathrm{km\,s^{-1}}$ (D) $6.25\ \mathrm{km\,s^{-1}}$

   ›Reveal solutionSolution

   A rod clamped at its middle vibrates with a node at the centre and antinodes at the free ends; for the fundamental mode the rod length equals half a wavelength, so $v = 2Lf = 5\ \mathrm{km\,s^{-1}}$.

   Concept & Intuition

   When a rod is clamped at its middle, the clamp forces that point to be a node (no displacement). The free ends can move freely, so they are antinodes (maximum displacement). For the fundamental (lowest-frequency) mode, the rod vibrates with a single “bulge” on each side of the clamp — that is, the rod’s length $L$ corresponds to exactly half a wavelength ($\lambda/2$) of the standing wave. The wave speed $v$ is related to frequency $f$ and wavelength $\lambda$ by $v = f\lambda$. Once we know $\lambda$, we can compute $v$.

   Step-by-step reasoning

   1. Identify the boundary conditions

      The rod is clamped at its centre → that point is a node. The two ends are free → they are antinodes. In the fundamental mode, the simplest standing wave that satisfies this has a node at the centre and antinodes at both ends.

   2. Relate rod length to wavelength

      From node to antinode is a quarter-wavelength ($\lambda/4$). The rod’s half-length (from centre to one end) is one quarter-wavelength. So the full length $L$ (200 cm) spans two quarter-wavelengths, i.e. half a wavelength:

   $L = \frac{\lambda}{2}$

   Hence

   $\lambda = 2L = 2 \times 200\ \mathrm{cm} = 400\ \mathrm{cm} = 4\ \mathrm{m}.$

   3. Apply the wave equation Speed $v = f\lambda$. Given $f = 1.25\ \mathrm{kHz} = 1250\ \mathrm{Hz}$ and $\lambda = 4\ \mathrm{m}$:

   $v = 1250 \times 4 = 5000\ \mathrm{m\,s^{-1}} = 5\ \mathrm{km\,s^{-1}}.$

   4. Match with the options …

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3. TG EAPCET 2026Set eng-2026-05-11-FN1 markMCQ
   Q.The lengths of two open pipes are in the ratio 5:6. If 8 beats are heard per second when both the pipes are vibrated in their fundamental modes, then the frequency of third harmonic of the longer pipe is (A) 144 Hz (B) 240 Hz (C) 120 Hz (D) 60 Hz

   ›Reveal solutionSolution

   The beat frequency equals the difference between the fundamental frequencies of the two open pipes. Using the length ratio and the beat condition, we find the fundamental of the longer pipe, then multiply by 3 to get its third harmonic. The answer is 120 Hz.

   Concept & Intuition

   For an open pipe, the fundamental frequency is $f = \frac{v}{2L}$, where $v$ is the speed of sound and $L$ is the length. When two pipes are sounded together, beats occur at the difference of their frequencies. Here the lengths are in ratio 5:6, so the shorter pipe has a higher fundamental. The beat frequency (8 Hz) gives us the difference. Once we find the fundamental of the longer pipe, its third harmonic is simply $3 \times$ that fundamental.

   Step-by-step solution

   1. Write the fundamental frequencies Let the lengths be $L_1 = 5x$ and $L_2 = 6x$ (ratio 5:6). For open pipes:

   $f_1 = \frac{v}{2L_1} = \frac{v}{10x}, \quad f_2 = \frac{v}{2L_2} = \frac{v}{12x}$

   Since $L_1 < L_2$, we have $f_1 > f_2$.

   2. Use the beat frequency Beats per second = $|f_1 - f_2| = 8$.

   $\frac{v}{10x} - \frac{v}{12x} = 8$

   Factor out $\frac{v}{x}$:

   $\frac{v}{x} \left( \frac{1}{10} - \frac{1}{12} \right) = 8$

   $\frac{v}{x} \cdot \frac{6 - 5}{60} = 8 \quad \Rightarrow \quad \frac{v}{x} \cdot \frac{1}{60} = 8$

   $\frac{v}{x} = 480$

   3. Find the fundamental of the longer pipe The longer pipe has length $L_2 = 6x$, so $$$$

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4. TG EAPCET 2026Set ap-2026-05-04-AN1 markMCQ
   Q.Second harmonic of an open pipe and third harmonic of a closed pipe of length 75 cm produce ‘n’ beats per second. If the fundamental frequency of the open pipe is 167 Hz, then the value of ‘n’ is (Speed of sound in air = $340~\mathrm{ms}^{-1}$) (A) 6 (B) 9 (C) 8 (D) 4

   ›Reveal solutionSolution

   We calculate the second harmonic frequency of the open pipe and the third harmonic frequency of the closed pipe (interpreted as the first overtone). The absolute difference between these two frequencies gives the beat frequency. The beat frequency is $\boxed{6}$.

   Concept and Intuition

   When sound waves are produced in pipes, they form standing waves at specific resonant frequencies, known as harmonics. The type of pipe (open at both ends or closed at one end) dictates which harmonics are allowed.

   • Open Pipe: An open pipe has antinodes (points of maximum displacement) at both ends. This boundary condition allows for all integer multiples of the fundamental frequency to be present. If $f_1$ is the fundamental frequency, then the allowed frequencies are $f_1, 2f_1, 3f_1, \dots, n f_1$. Here, $n$ represents the $n$-th harmonic. The fundamental frequency for an open pipe of length $L$ is $f_1 = \frac{v}{2L}$, where $v$ is the speed of sound.

   • Closed Pipe: A closed pipe has a node (point of zero displacement) at the closed end and an antinode at the open end. This boundary condition restricts the allowed frequencies to only odd integer multiples of the fundamental frequency. If $f_1'$ is the fundamental frequency, then the allowed frequencies are $f_1', 3f_1', 5f_1', \dots, (2k-1)f_1'$. Here, $f_1'$ is the 1st harmonic, $3f_1'$ is the 3rd harmonic (or 1st overtone), $5f_1'$ is the 5th harmonic (or 2nd overtone), and so on. The fundamental frequency for a closed pipe of length $L$ is $f_1' = \frac{v}{4L}$.

   • Beat Frequency: When two sound waves of slightly different frequencies ($f_A$ and $f_B$) interfere, they produce a phenomenon called beats. The listener perceives a periodic variation in the loudness of the sound. The number of beats heard per second is called the beat frequency, which is simply the absolute difference between the two frequencies: $f_{beat} = |f_A - f_B|$.

   Step-by-step Solution

   1. Determine the frequency of the second harmonic of the open pipe.

      For an open pipe, the frequency of the $n$-th harmonic is given by $f_n = n f_1$, where $f_1$ is the fundamental frequency.

      We are given the fundamental frequency of the open pipe, $f_{1, \text{open}} = 167~\mathrm{Hz}$.

      The second harmonic frequency of the open pipe is:

      $f_{2, \text{open}} = 2 \times f_{1, \text{open}}$

      $f_{2, \text{open}} = 2 \times 167~\mathrm{Hz}$

      $f_{2, \text{open}} = 334~\mathrm{Hz}$

   2. Determine the frequency of the third harmonic of the closed pipe.

      First, let's find the fundamental frequency of the closed pipe. The formula for the fundamental frequency of a closed pipe of length $L$ is:

      $f_{1, \text{closed}} = \frac{v}{4L}$

      We are given the speed of sound $v = 340~\mathrm{ms}^{-1}$ and the length of the pipe $L = 75~\mathrm{cm} = 0.75~\mathrm{m}$.

      Substitute these values:

      $f_{1, \text{closed}} = \frac{340~\mathrm{ms}^{-1}}{4 \times 0.75~\mathrm{m}}$

      $f_{1, \text{closed}} = \frac{340}{3}~\mathrm{Hz}$

      Now, we need to find the "third harmonic" of the closed pipe. This term can sometimes be ambiguous. …

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5. TG EAPCET 2026Set ap-2026-05-04-FN1 markMCQ
   Q.The frequency of a closed pipe is $f_1$ if it vibrates with two nodes and is $f_2$ if it vibrates with three nodes. If the difference between the frequencies $f_1$ and $f_2$ is $200~\mathrm{Hz}$, then the length of the pipe is (Speed of sound in air $= 340~\mathrm{ms}^{-1}$) (A) 75 cm (B) 65 cm (C) 85 cm (D) 55 cm

   ›Reveal solutionSolution

   We determine the harmonic numbers corresponding to two and three nodes in a closed pipe, then use the given frequency difference to calculate the pipe's length. The length of the pipe is $\boxed{85~\mathrm{cm}}$.

   Concept and Intuition

   When sound waves are produced in a pipe, they reflect from the ends, creating standing waves. For a closed pipe, the air molecules at the closed end cannot move, forming a node (a point of zero displacement). At the open end, the air molecules can move freely, forming an antinode (a point of maximum displacement).

   The key idea is that a standing wave in a closed pipe must always have a node at the closed end and an antinode at the open end. The distance between a consecutive node and antinode is always $\lambda/4$, where $\lambda$ is the wavelength of the sound wave. The distance between two consecutive nodes (or two consecutive antinodes) is $\lambda/2$.

   The possible standing wave patterns in a closed pipe are such that the length of the pipe $L$ must be an odd multiple of $\lambda/4$. That is, $L = (2n-1)\frac{\lambda}{4}$, where $n=1, 2, 3, \dots$.

   The corresponding frequencies are given by $f = \frac{v}{\lambda}$, where $v$ is the speed of sound. Substituting $\lambda = \frac{4L}{2n-1}$, we get:

   The frequency of the $n$-th harmonic in a closed pipe is $f_n = \frac{(2n-1)v}{4L}$, where $n=1, 2, 3, \dots$.

   These are the fundamental (1st harmonic), 3rd harmonic, 5th harmonic, and so on. Only odd harmonics are present in a closed pipe.

   Let's relate the number of nodes to the harmonic number:

   • 1st harmonic (fundamental): Has 1 node (at the closed end) and 1 antinode (at the open end). $L = \lambda/4$.
   • 3rd harmonic (first overtone): Has 2 nodes and 2 antinodes. The pattern is Node-Antinode-Node-Antinode. $L = 3\lambda/4$.
   • 5th harmonic (second overtone): Has 3 nodes and 3 antinodes. The pattern is Node-Antinode-Node-Antinode-Node-Antinode. $L = 5\lambda/4$.

   In general, for the $(2n-1)$-th harmonic, there are $n$ nodes and $n$ antinodes.

   Step-by-step Derivation

   1. Identify the harmonic for $f_1$ (two nodes):

      A closed pipe vibrating with two nodes means it has the pattern Node-Antinode-Node-Antinode.

      The length of the pipe $L$ for this pattern is the distance from the first node to the last antinode.

      $L = \frac{\lambda}{4} + \frac{\lambda}{2} = \frac{3\lambda}{4}$.

      This corresponds to the 3rd harmonic.

      Therefore, the frequency $f_1$ is given by:

      $f_1 = \frac{3v}{4L}$

   2. Identify the harmonic for $f_2$ (three nodes):

      A closed pipe vibrating with three nodes means it has the pattern Node-Antinode-Node-Antinode-Node-Antinode. …

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6. TG EAPCET 2025Set eng-2025-05-02-FN1 markMCQ
   Q.For commercial telephonic communication, the frequency range adequate for speech signals is (A) 20 Hz - 20 kHz (B) 300 Hz - 3100 Hz (C) 200 MHz - 600 MHz (D) 300 kHz - 8000 kHz

   ›Reveal solutionSolution

   The key idea is that commercial telephony only needs to transmit intelligible speech, not full audio fidelity, so the frequency range is deliberately narrowed to save bandwidth. The correct range is 300 Hz – 3100 Hz, which corresponds to option (B).

   The question asks about the frequency range used for commercial telephonic communication — that is, the range of frequencies that a standard telephone system (like the Public Switched Telephone Network, or PSTN) actually transmits for speech. This is not about the full range of human hearing or the theoretical range of speech, but about the practical, engineered bandwidth chosen to balance voice clarity with efficient use of the transmission channel.

   Why this approach works:

   Human speech contains frequencies from about 100 Hz to 8 kHz, but most of the energy and intelligibility (especially for consonants) lies between roughly 300 Hz and 3.4 kHz. Telephone engineers discovered that cutting off frequencies below 300 Hz (which carry mostly low-frequency hum and some vowel power) and above 3.4 kHz (which carry sibilants but are less critical for understanding) still yields perfectly understandable speech. This narrow band (about 3.1 kHz wide) allows many calls to be multiplexed onto a single line, saving enormous cost and infrastructure.

   Let’s walk through the options:

   1. Option (A): 20 Hz – 20 kHz

      This is the full range of human hearing — used for high-fidelity music, not telephony. Transmitting this would require far too much bandwidth and is unnecessary for speech intelligibility. So this is incorrect.

   2. Option (B): 300 Hz – 3100 Hz

      This matches the standard telephone voice band. The lower cutoff at 300 Hz removes low-frequency noise and mains hum (50/60 Hz), while the upper cutoff at 3100 Hz (often given as 3.4 kHz in practice) preserves enough high-frequency content to distinguish consonants like “s” and “f”. This is the correct answer.

   3. Option (C): 200 MHz – 600 MHz …

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7. TG EAPCET 2025Set eng-2025-05-02-AN1 markMCQ
   Q.If the lengths of the open and closed pipes are in the ratio of $2:3$, then the ratio of the frequencies of the third harmonic of the open pipe and the fifth harmonic of the closed pipe is (A) $3:5$ (B) $9:5$ (C) $2:3$ (D) $4:9$

   ›Reveal solutionSolution

   The ratio of the third harmonic of an open pipe to the fifth harmonic of a closed pipe, given their lengths in the ratio 2:3, is 9:5 — corresponding to option (B).

   Concept & Intuition

   The key idea is that open and closed pipes produce different harmonic series. An open pipe supports all harmonics (fundamental, 2nd, 3rd, …), while a closed pipe supports only odd harmonics (1st, 3rd, 5th, …). The frequency of a given harmonic depends on the speed of sound (same for both), the length of the pipe, and the harmonic number. By writing the frequency formulas, substituting the given length ratio, and simplifying, we get the required ratio.

   Step-by-step solution

   1. Recall the frequency formulas For an open pipe of length $L_o$, the frequency of the $n$th harmonic is

   $f_o^{(n)} = \frac{n v}{2 L_o}$

   where $v$ is the speed of sound.

   For a closed pipe of length $L_c$, only odd harmonics exist; the frequency of the $m$th odd harmonic (where $m = 1,3,5,\dots$) is

   $f_c^{(m)} = \frac{m v}{4 L_c}$

   2. Identify the specific harmonics The third harmonic of the open pipe corresponds to $n = 3$:

   $f_o^{(3)} = \frac{3 v}{2 L_o}$

   The fifth harmonic of the closed pipe corresponds to $m = 5$:

   $f_c^{(5)} = \frac{5 v}{4 L_c}$

   3. Use the given length ratio The lengths are in the ratio $L_o : L_c = 2 : 3$. So we can write

   $L_o = 2k, \quad L_c = 3k$

   for some positive constant $k$.

   4. Write the ratio of frequencies $$$$

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8. TG EAPCET 2025Set eng-2025-05-04-AN1 markMCQ
   Q.The air columns in two tubes closed at one end vibrating in their fundamental modes produce 2 beats per second. The number of beats produced per second when the same tubes are vibrated in their fundamental mode with their both ends open are (A) 1 (B) 2 (C) 3 (D) 4

   ›Reveal solutionSolution

   The beat frequency for closed tubes is proportional to the difference in their fundamental frequencies; for open tubes, the fundamental frequencies double, so the beat frequency also doubles — from 2 beats/s to 4 beats/s.

   ---

   Concept & Intuition

   Beats occur when two sound waves of slightly different frequencies interfere. The beat frequency is simply the absolute difference of the two frequencies.

   For a tube closed at one end, the fundamental frequency is $f_{\text{closed}} = \frac{v}{4L}$, where $v$ is the speed of sound and $L$ is the tube length.

   For a tube open at both ends, the fundamental frequency is $f_{\text{open}} = \frac{v}{2L}$.

   Notice that $f_{\text{open}} = 2 f_{\text{closed}}$. So if two closed tubes have frequencies $f_1$ and $f_2$, their open-tube frequencies become $2f_1$ and $2f_2$. The difference then doubles, and so does the beat frequency.

   ---

   Step-by-step reasoning

   1. Let the two closed tubes have lengths $L_1$ and $L_2$. Their fundamental frequencies (closed at one end) are:

   $f_1 = \frac{v}{4L_1}, \quad f_2 = \frac{v}{4L_2}.$

   2. Given: they produce 2 beats per second in their fundamental modes.

      Beat frequency = $|f_1 - f_2| = 2$ Hz.

   3. Now consider the same tubes with both ends open.

      The fundamental frequency for an open tube is:

   $f_1' = \frac{v}{2L_1} = 2 \cdot \frac{v}{4L_1} = 2f_1,$

   $$$$

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9. TG EAPCET 2025Set eng-2025-05-04-FN1 markMCQ
   Q.The air columns in two tubes closed at one end vibrating in their fundamental modes produce 2 beats per second. The number of beats produced per second when the same tubes are vibrated in their fundamental mode with their both ends open are (A) 1 (B) 2 (C) 4 (D) 3

   ›Reveal solutionSolution

   When two air columns closed at one end produce 2 beats/second, opening both ends of the tubes doubles their fundamental frequencies, leading to a doubling of the beat frequency. The new beat frequency is $\boxed{4}$ beats per second.

   The problem asks us to find the beat frequency when two air columns, initially closed at one end, are opened at both ends. The key to solving this lies in understanding how the fundamental frequency of an air column changes when its boundary conditions (closed vs. open ends) are altered.

   Concept and Intuition

   1. Fundamental Frequency of Air Columns:

      • Tube closed at one end: For a tube closed at one end, the fundamental mode of vibration has a node at the closed end and an antinode at the open end. This means the length of the tube $L$ corresponds to one-quarter of the wavelength ($\lambda/4$).

        The fundamental frequency for a tube closed at one end is $f_c = \frac{v}{4L}$, where $v$ is the speed of sound in air and $L$ is the length of the tube.

      • Tube open at both ends: For a tube open at both ends, the fundamental mode of vibration has antinodes at both ends and a node in the middle. This means the length of the tube $L$ corresponds to one-half of the wavelength ($\lambda/2$).

        The fundamental frequency for a tube open at both ends is $f_o = \frac{v}{2L}$, where $v$ is the speed of sound in air and $L$ is the length of the tube.

   2. Relationship between $f_c$ and $f_o$:

      Comparing the two formulas, we can see a direct relationship:

      $f_o = \frac{v}{2L} = 2 \times \left(\frac{v}{4L}\right) = 2f_c$.

      This is a crucial insight: for a given tube, its fundamental frequency when open at both ends is exactly twice its fundamental frequency when closed at one end.

   3. Beat Frequency:

      When two sound waves of slightly different frequencies $f_1$ and $f_2$ interfere, they produce beats. The beat frequency is the absolute difference between their frequencies.

      Beat frequency $f_{beat} = |f_1 - f_2|$.

   Step-by-step Derivation

   1. Frequencies of the closed tubes:

      Let the two tubes have lengths $L_1$ and $L_2$.

      When closed at one end, their fundamental frequencies are:

      $f_{c1} = \frac{v}{4L_1}$

      $f_{c2} = \frac{v}{4L_2}$

   2. Beat frequency for closed tubes:

      We are given that these tubes produce 2 beats per second when vibrating in their fundamental modes.

      So, $|f_{c1} - f_{c2}| = 2 \text{ Hz}$. …

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10. TG EAPCET 2025Set ap-2025-04-29-AN1 markMCQ
    Q.The successive frequencies produced by an organ pipe are 330 Hz, 440 Hz and 550 Hz. If the speed of sound in air is $330\,\mathrm{ms}^{-1}$, then the length of the organ pipe is (A) 3 m (B) 1.5 m (C) 2 m (D) 0.75 m

    ›Reveal solutionSolution

    The given frequencies are in arithmetic progression with a common difference of 110 Hz, which is the fundamental frequency of an open pipe. Using $f_1 = v/2L$, the length is $L = 1.5$ m. The correct option is (B).

    The key is to recognise what kind of organ pipe produces successive frequencies that are equally spaced. For an open organ pipe, the natural frequencies are $f_n = n \cdot \frac{v}{2L}$, where $n = 1, 2, 3, \dots$ — these are integer multiples of a fundamental, so they form an arithmetic progression with a common difference equal to the fundamental frequency itself. For a closed organ pipe, only odd harmonics are present ($f_n = (2n-1)\frac{v}{4L}$), so successive frequencies are not equally spaced in the same simple way. Here, 330 Hz, 440 Hz, and 550 Hz differ by exactly 110 Hz each — a clear arithmetic progression. That tells us the pipe is open at both ends, and the common difference (110 Hz) is the fundamental frequency.

    1. Identify the pipe type.

       The three frequencies are $330$, $440$, and $550$ Hz. Their differences: $440 - 330 = 110$ Hz, $550 - 440 = 110$ Hz. Since they are equally spaced, the pipe must be open. For an open pipe, the $n$th harmonic is $f_n = n f_1$, where $f_1$ is the fundamental. Here $f_1 = 110$ Hz.

    2. Check consistency with the given numbers.

       If $f_1 = 110$ Hz, then $f_2 = 220$ Hz, $f_3 = 330$ Hz, $f_4 = 440$ Hz, $f_5 = 550$ Hz. The given frequencies correspond to the 3rd, 4th, and 5th harmonics of an open pipe. That fits perfectly — no missing or extra frequencies in between.

    3. Apply the formula for an open pipe.

       For an open organ pipe, the fundamental frequency is

       $f_1 = \frac{v}{2L}$ …

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11. TG EAPCET 2025Set ap-2025-04-29-FN1 markMCQ
    Q.The distance between the objective and eyepiece of an astronomical telescope when the final image forms at infinity is 62 cm. If the magnification of the telescope is 30, the focal lengths of the objective and eyepiece respectively are (A) 31 cm, 2 cm (B) 2 cm, 31 cm (C) 2 cm, 60 cm (D) 60 cm, 2 cm

    ›Reveal solutionSolution

    For an astronomical telescope in normal adjustment (final image at infinity), the tube length equals $f_o + f_e$ and the magnifying power equals $f_o / f_e$. Solving these two equations gives $f_o = 60$ cm and $f_e = 2$ cm.

    The key idea is that when the final image is at infinity — called normal adjustment — the telescope is set so that the objective forms its image at the focus of the eyepiece. This makes the distance between the two lenses exactly the sum of their focal lengths. The magnification in this setting is simply the ratio of the focal lengths. So we have two equations in two unknowns, and solving them is straightforward.

    1. Write down the two conditions. Let $f_o$ be the focal length of the objective and $f_e$ that of the eyepiece.
       • Tube length (distance between lenses) when final image is at infinity:

    $L = f_o + f_e = 62 \text{ cm}$

    • Magnifying power (magnitude) for normal adjustment:

    $M = \frac{f_o}{f_e} = 30$

    2. Solve for $f_o$ and $f_e$. From the second equation, $f_o = 30 f_e$. Substitute into the first:

    $30 f_e + f_e = 62 \quad\Rightarrow\quad 31 f_e = 62 \quad\Rightarrow\quad f_e = 2 \text{ cm}$

    Then $f_o = 30 \times 2 = 60 \text{ cm}$.

    3. Check the options. The pair (60 cm, 2 cm) matches option (D). …

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12. TG EAPCET 2024Set ap-2024-05-07-AN1 markMCQ
    Q.A and B are two points of a string, in which a standing wave of wavelength $\lambda$ is set up. If the distance between the points A and B is $\dfrac{3\lambda}{4}$, then the phase difference between A and B is (A) $\dfrac{\pi}{3}$ (B) $\dfrac{3\pi}{4}$ (C) $\dfrac{3\pi}{2}$ (D) $\pi$

    ›Reveal solutionSolution

    The phase difference between two points in a wave is directly proportional to the path difference between them. For a path difference of $\frac{3\lambda}{4}$, the phase difference is $\boxed{\frac{3\pi}{2}}$.

    Concept and Intuition

    A wave is a disturbance that propagates through a medium, and its phase describes the state of oscillation at a particular point and time. The phase of a wave changes as we move along its path. A complete cycle of the wave, which spans one wavelength ($\lambda$), corresponds to a full $2\pi$ radians (or $360^\circ$) change in phase. This fundamental relationship between path difference ($\Delta x$) and phase difference ($\Delta \phi$) is crucial for all types of waves.

    While standing waves have unique characteristics, such as fixed nodes (points of zero displacement) and antinodes (points of maximum displacement), and particles between consecutive nodes oscillating in phase (or out of phase by $\pi$ across a node), the underlying relationship between a physical distance and the phase change it represents in terms of the wave's propagation constant ($k = 2\pi/\lambda$) remains consistent. In competitive exams, when "phase difference" is asked for a given path difference in a wave (whether progressive or standing), the general formula derived from progressive waves is almost always the intended method. It represents the phase difference that would accumulate over that distance.

    Step-by-step Derivation

    1. Identify the given quantities:

       We are given that the wavelength of the standing wave is $\lambda$.

       The distance between points A and B is $\Delta x = \frac{3\lambda}{4}$.

    2. Recall the formula for phase difference:

       The phase difference $\Delta \phi$ between two points separated by a path difference $\Delta x$ in any wave with wavelength $\lambda$ is given by the formula:

       $\Delta \phi = \frac{2\pi}{\lambda} \Delta x$

    3. Substitute the given values into the formula: …

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