Questions & explanations
1. Two tuning forks A and B produce 4 beats per second when sounded together. The frequency of fork A is 256 Hz. When fork A is loaded with a little wax, the beat frequency reduces to 2 beats per second. What is the frequency of fork B?
- 250 Hz
- 252 Hz
- 260 Hz
- 262 Hz
Answer: 252 Hz
Initially, the beat frequency of 4 Hz means that the difference between the frequencies of A and B is 4 Hz. So fork B could be either 256 + 4 = 260 Hz or 256 − 4 = 252 Hz. Loading fork A with wax decreases its frequency (mass increases, frequency decreases). The beat frequency decreases to 2 Hz, which means the two frequencies are now closer together. This is only possible if fork B had a lower frequency than A originally; if B were higher, decreasing A would increase the difference. Hence B is 252 Hz.
2. A tuning fork of frequency 512 Hz is set into vibration and brought near a stretched string that is vibrating. The string vibrates with maximum amplitude when its frequency is exactly 512 Hz. This phenomenon is called:
- Beats
- Resonance
- Free oscillation
- Damped oscillation
Answer: Resonance
Resonance occurs when the frequency of an external driving force (here the vibrating tuning fork) matches the natural frequency of the receiving system (the string). At this matching frequency, maximum energy transfer takes place, resulting in the largest amplitude of vibration. Beats arise from the superposition of two slightly different frequencies. Free oscillation occurs without any external periodic force. Damped oscillation involves a gradual decrease in amplitude due to energy loss.
3. Two points on a sinusoidal progressive wave are separated by a distance of 2.5 m. If the wavelength of the wave is 10 m, the phase difference between the two points is:
- π/4 rad
- π/2 rad
- π rad
- 0 rad
Answer: π/2 rad
The phase difference Δφ is related to the path difference Δx by Δφ = (2π/λ) × Δx. Here Δx = 2.5 m and λ = 10 m, so Δφ = (2π/10) × 2.5 = (2π × 0.25) = π/2 rad. Option A (π/4 rad) would correspond to a path difference of 1.25 m; option C (π rad) corresponds to 5 m; option D (0 rad) corresponds to zero path difference or an integer multiple of λ (10 m, 20 m, etc.). Only option B gives the correct phase difference for the given separation and wavelength.
4. Which of the following correctly identifies the type of wave and its example?
- Transverse – Sound wave in air
- Longitudinal – Light wave
- Transverse – Wave on a stretched string
- Longitudinal – Water wave in deep water
Answer: Transverse – Wave on a stretched string
A wave on a stretched string is transverse because particles vibrate perpendicular to the direction of wave propagation. Sound is a longitudinal wave, light is a transverse wave, and deep-water waves are a combination, not purely longitudinal.
5. Which of the following statements is true for the harmonics produced in an open organ pipe compared to a closed organ pipe of the same length?
- Open pipe produces all harmonics, closed pipe produces only even harmonics.
- Open pipe produces all harmonics, closed pipe produces only odd harmonics.
- Open pipe produces only odd harmonics, closed pipe produces all harmonics.
- Both produce all harmonics.
Answer: Open pipe produces all harmonics, closed pipe produces only odd harmonics.
An open organ pipe supports both odd and even harmonics (all integer multiples of the fundamental). A closed organ pipe allows only odd harmonics because the closed end imposes a node, restricting the possible standing wave patterns.
6. Laplace correction in the formula for the speed of sound in a gas accounts for the fact that the propagation of sound is:
- Isothermal
- Adiabatic
- Isochoric
- Isobaric
Answer: Adiabatic
Newton originally assumed isothermal compression and rarefaction. Laplace corrected it by pointing out that compressions and rarefactions occur so rapidly that heat exchange is negligible, i.e., the process is adiabatic.
7. According to the principle of superposition of waves, when two waves meet at a point, the resultant displacement is:
- the algebraic sum of the individual displacements
- the product of the individual displacements
- the average of the individual displacements
- the difference of the individual displacements
Answer: the algebraic sum of the individual displacements
The principle of superposition states that when two or more waves overlap, the resultant displacement at any point is the vector sum (or algebraic sum for collinear waves) of the individual displacements.
8. A stretched string fixed at both ends vibrates in its fundamental mode. If the length of the string is L and the speed of transverse waves on it is v, the fundamental frequency is:
- v/(2L)
- v/L
- 2v/L
- v/(4L)
Answer: v/(2L)
For a string fixed at both ends, the fundamental mode has a node at each end and an antinode in the middle, so the length L equals λ/2. Hence λ = 2L, and frequency f = v/λ = v/(2L).
9. For an organ pipe closed at one end, the length of the pipe L is related to the wavelength λ of the fundamental mode as:
- L = λ/2
- L = λ/4
- L = λ
- L = 2λ
Answer: L = λ/4
In a closed organ pipe, the fundamental mode has a node at the closed end and an antinode at the open end, so the length L corresponds to one-quarter of the wavelength (L = λ/4).
10. For constructive interference of two coherent waves of the same wavelength λ, the path difference between them must be:
- (2n+1)λ/2
- nλ
- (2n+1)λ/4
- nλ/2
Answer: nλ
Constructive interference occurs when the waves arrive in phase, which requires a path difference of an integer multiple of the wavelength, i.e., nλ (where n = 0, 1, 2, ...).
11. The speed of sound in air at 27°C is 340 m/s. What will be its speed at 327°C?
- 340 m/s
- 680 m/s
- 480 m/s
- 600 m/s
Answer: 480 m/s
Speed of sound in a gas is proportional to √T (in Kelvin). T₁ = 27°C = 300 K, T₂ = 327°C = 600 K. v₂ = v₁ × √(T₂/T₁) = 340 × √2 ≈ 340 × 1.414 = 480 m/s.
12. In a stationary wave formed by the superposition of two identical waves traveling in opposite directions, the points of zero displacement are called:
- antinodes
- nodes
- crests
- troughs
Answer: nodes
Nodes are points in a standing wave where the displacement is always zero due to destructive interference. Antinodes are points of maximum displacement.