Waves

Step 1: Conservation of momentum.
Since the bullet remains embedded in the block, this is an inelastic collision. The total momentum before and after the collision must be conserved. The initial momentum of the system is , and the final momentum is . Thus, by conservation of momentum: Solving for , the final velocity of the system:
Step 2: Conclusion.
Thus, the correct answer is (D) .

Step 1: Magnifying power of telescope.
The magnifying power of a telescope is given by the ratio of the focal length of the objective to the focal length of the eyepiece: For maximum magnifying power, the objective should have a large focal length and the eyepiece should have a small focal length.
Step 2: Conclusion.
Thus, the correct answer is (A) large and small focal length.

Step 1: Understanding amplitude modulation.
In amplitude modulation (AM), the amplitude of the carrier wave varies according to the information signal, while the frequency remains constant.
Step 2: Conclusion.
Thus, the correct answer is (D) amplitude of carrier wave changes according to information signal.

Step 1: Formula for capacitance.
The capacitance with a dielectric is given by: where is the capacitance without the dielectric and is the dielectric constant. We are given that and , so:
Step 2: Finding the permittivity of the medium.
The permittivity is related to and by: Substituting the values:
Step 3: Conclusion.
The correct answer is (C) .

Step 1: Identify wave number.
From the given equation:
The wave number is:


Step 2: Relation between wavelength and wave number.


Step 3: Distance between consecutive nodes.
Distance between two consecutive nodes is .


Step 4: Conclusion.
The distance between two consecutive nodes is .

Step 1: Expression for fundamental frequency of a stretched string.
where is mass per unit length.

Step 2: Mass per unit length relation.
For same material, is constant.

Step 3: Calculate for both strings.
First string:

Second string:


Step 4: Frequency ratio.


Step 5: Conclusion.
The ratio of fundamental frequencies is .

Step 1: Understanding direct reading instruments.
Direct reading instruments give the value of the quantity being measured directly on a scale or display without requiring additional calculations. Examples include voltmeters, ammeters, and electronic balances.
Step 2: Potentiometer behavior.
A potentiometer is not a direct reading instrument. It is used to compare voltages, and it requires an additional calculation to find the voltage of the unknown source.
Step 3: Conclusion.
The correct answer is (D) Potentiometer, as it is not a direct reading instrument.

Step 1: Understanding the photoelectric effect.
The energy of the incoming photon is , where is Planck's constant, is the speed of light, and is the wavelength of the incident light. The work function of the metal is , and the energy required to emit an electron is equal to the energy of the photon minus the work function.
Step 2: Maximum velocity of the electron.
The maximum kinetic energy of the emitted electron is given by the equation: Solving for the velocity , we get:
Step 3: Conclusion.
The maximum velocity of the emitted electron is , so the correct answer is (B).

Step 1: First overtone frequencies.
For closed organ pipe, first overtone corresponds to 3rd harmonic:
For open organ pipe, first overtone corresponds to 2nd harmonic:

Step 2: Equate frequencies.


Step 3: Find ratio of lengths.

Step 1: Relation between tension and number of loops.
In Melde’s experiment,

Step 2: Write ratio of tensions.
Let initial tension be for 4 loops and final tension be for 5 loops.


Step 3: Use given decrease in tension.

Solving these equations,

Step 1: Formation of stationary waves.
A stationary wave is formed by the superposition of two identical progressive waves having the same amplitude, frequency, and wavelength but traveling in opposite directions.

Step 2: Wavelength relation.
The wavelength of the stationary wave is equal to the wavelength of the individual progressive waves.

Step 3: Interpretation of nodes and antinodes.
In a stationary wave, the distance between two consecutive nodes or antinodes is , but the full wavelength remains .

Step 4: Final conclusion.
Hence, the wavelength of the stationary wave is .

Step 1: Doppler effect formula for moving observer.
where is observer’s velocity towards the source.

Step 2: Condition for doubled frequency.


Step 3: Solve for observer’s velocity.

Step 1: Understanding the resonance condition.
The pipe closed at one end has a fundamental frequency with a wavelength of , where is the length of the pipe. The string vibrating in the second harmonic must have a wavelength that matches this resonance condition. The wavelength of the string in the second harmonic is .
Step 2: Using the wave equation.
The mass per unit length of the string is , and the wave speed on the string is given by: Substituting the given values and solving for the mass, we find that the mass of the string is 10 grams.
Step 3: Conclusion.
The correct answer is (C), 10 grams.

Step 1: Using the wave speed formula.
The speed of transverse wave along a string is given by the formula: where is the tension and is the mass per unit length, which is given by , where is the area of cross-section.
Step 2: Solving for .
By substituting into the wave speed formula, we get: Step 3: Conclusion.
The correct answer is (B), .

Step 1: Formula for resonant frequencies of an open pipe.
For an open pipe:


Step 2: Calculate fundamental frequency.


Step 3: Possible resonant frequencies.


Step 4: Identify non-resonant frequency.
85 Hz is not an integral multiple of 170 Hz.

Step 5: Conclusion.
The pipe cannot resonate at 85 Hz.

Step 1: Identify the given data.
Velocity of bus (source and observer),
Original frequency of horn,
Speed of sound,
The bus is moving towards a stationary wall, so Doppler effect occurs twice (during incidence and reflection).

Step 2: Frequency of sound reaching the wall.
Since the source (bus) is moving towards the wall, the frequency heard by the wall is:


Step 3: Frequency of reflected sound heard by passengers.
Now the reflected sound acts as a source from the wall, and the passengers (observer) are moving towards it:


Step 4: Calculate number of beats per second.
Beats per second


Step 5: Conclusion.
The number of beats heard per second by the passengers is 5.

Step 1: Understanding the relationship between wave velocity and stress.
The wave velocity on a stretched string is given by: Where: - is the tensile stress, - is the density. Rearranging the equation for stress , we get: Thus, the correct answer is (D) .

Step 1: Resonance condition.
For resonance to occur, the length of the resonance tube must satisfy the condition where the tube supports standing waves. The formula for the frequency of resonance is: Where: - - - Step 2: Calculating the number of resonances.
The total number of resonances can be found by: Substituting the known values: Thus, the correct answer is (B) 4.

Step 1: Doppler effect when source moves away.
Given:

Step 2: Finding velocity ratio.

Step 3: Observer moving away, source at rest.

Step 4: Ratio of frequencies.

Step 5: Conclusion.
The required ratio is .

Step 1: Understanding the harmonics.
For an open organ pipe, the harmonics are given by: where is the frequency of the -th harmonic, and is the fundamental frequency. The third harmonic is given by:
Step 2: Using the given information.
We are told that the third harmonic frequency is 50 Hz more than the fundamental frequency: Substituting , we get: Solving for , we get:
Step 3: Conclusion.
The fundamental frequency is 100 Hz, which corresponds to option (B).

Step 1: Relationship between frequency and tension.
The frequency of a vibrating wire is proportional to the square root of the tension, i.e., Let the initial frequency be and the initial tension be . If the tension is increased by 2%, the new frequency will be , and we have: This gives a new frequency . The beat frequency is given by the difference between the two frequencies: We are given that the beat frequency is 5 Hz. Therefore,
Step 2: Solving for the initial frequency.
Using the approximation , we get:
Step 3: Conclusion.
The initial frequency of each wire is 500 Hz, which corresponds to option (B).

Step 1: Understanding the setup.
The string is vibrating in its 1st overtone, which corresponds to the fundamental frequency of the pipe. The pipe’s fundamental frequency and the string’s frequency are related. The string's fundamental frequency can be expressed as: where is the speed of the wave on the string, and is the length of the string.

Step 2: Applying the fundamental frequency relationship.
The frequency of the pipe is related to its length and the speed of sound in air. The fundamental frequency of the pipe is given by: where , and . Using this, we find the frequency of the pipe.
Step 3: Solving for the mass of the string.
Using the relationship between frequency, tension, and mass per unit length of the string, we solve for the mass of the string and find it to be 10 grams.

Step 1: Understanding the fundamental mode of a pipe.
In the fundamental mode of vibration in a pipe closed at one end, the pipe behaves as a quarter-wavelength resonator. The length of the pipe is related to the wavelength of the sound wave by: The speed of sound is related to the frequency and the wavelength by: Since , we substitute this into the equation: Step 2: Solving for the frequency.
Given that the time is the time it takes for the sound wave to travel from the open end to the closed end, the frequency of the vibration is the inverse of the time for one complete oscillation, which gives .
Step 3: Conclusion.
Thus, the correct answer is (C) .

Step 1: Understanding the modes of vibration.
In a pipe closed at one end, the fundamental mode of vibration has one node at the closed end and one antinode at the open end. The first overtone (or second harmonic) has one node and one antinode inside the pipe, and the second overtone (third harmonic) has two nodes and two antinodes. Thus, when there are 2 nodes and 2 antinodes, the vibration corresponds to the first overtone.
Step 2: Conclusion.
Thus, the correct answer is (B) 1st overtone.