Waves

A wave travels along a string of linear mass density \(0.05\ \text{kg/m}\) under tension 20 N. The wave speed is:

1. a\(40\ \text{m/s}\)
2. b\(10\ \text{m/s}\)
3. c\(5\ \text{m/s}\)
4. d\(20\ \text{m/s}\)

The frequency of a wave is 500 Hz and its wavelength is 0.6 m. The wave speed is:

1. a\(100\ \text{m/s}\)
2. b\(300\ \text{m/s}\)
3. c\(500\ \text{m/s}\)
4. d\(60\ \text{m/s}\)

Speed of sound in air at 0 °C is \(330\ \text{m/s}\). At 27 °C, the speed is approximately:

1. a\(330\ \text{m/s}\)
2. b\(320\ \text{m/s}\)
3. c\(346\ \text{m/s}\)
4. d\(360\ \text{m/s}\)

Two sound sources have frequencies 300 Hz and 304 Hz. The number of beats heard per second is:

1. a604
2. b2
3. c8
4. d4

A pipe closed at one end has fundamental frequency 200 Hz. The next allowed frequency (first overtone) is:

1. a400 Hz
2. b600 Hz
3. c200 Hz
4. d1000 Hz

A pipe open at both ends of length 1 m has fundamental frequency:

1. a\(\dfrac{v}{4}\) (Hz)
2. b\(\dfrac{v}{2}\) (Hz)
3. c\(\dfrac{v}{L} = v\) Hz
4. d\(v\) Hz

Two stretched strings of the same material and same cross-section have lengths in the ratio 1 : 2 and carry the same tension. The ratio of their fundamental frequencies \(f_{1} : f_{2}\) is:

1. a4 : 1
2. b1 : 2
3. c2 : 1
4. d1 : 4

A standing wave on a string fixed at both ends has 4 antinodes between the ends. The wavelength of the standing wave on a 1 m string is:

1. a2 m
2. b0.25 m
3. c1 m
4. d0.5 m

A wave is described by \(y(x, t) = 0.02\sin(8x - 320t)\) where \(x\) is in m, \(t\) in s. The wave speed is:

1. a\(320\ \text{m/s}\)
2. b\(40\ \text{m/s}\)
3. c\(8\ \text{m/s}\)
4. d\(1280\ \text{m/s}\)

Two waves of equal amplitude \(A\) and equal frequency, but with phase difference \(\pi\), superpose. The amplitude of the resultant is:

1. a\(0\)
2. b\(A\)
3. c\(2A\)
4. d\(A \cdot \sqrt{2}\)

If the tension in a string is increased 4-fold, the speed of a transverse wave on it changes by a factor of:

1. a\(\dfrac{1}{2}\)
2. b\(4\)
3. c\(2\)
4. d\(16\)

A taut string of length \(L\) vibrates in its third harmonic with frequency 600 Hz. The fundamental frequency is:

1. a1800 Hz
2. b300 Hz
3. c100 Hz
4. d200 Hz

Assertion (A): Increasing the tension in a string increases the speed of a transverse wave on it.

Reason (R): The speed of a transverse wave on a stretched string is given by \(v = \sqrt{T/\mu}\), where \(T\) is the tension and \(\mu\) is the linear mass density.

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

Assertion (A): Sound cannot travel through vacuum.

Reason (R): Sound is a longitudinal mechanical wave that requires a material medium for its propagation.

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

Assertion (A): Newton's formula \(v = \sqrt{P/\rho}\) gives the speed of sound in air at STP as about 280 m/s, roughly 15% less than the measured value of about 331 m/s.

Reason (R): Newton assumed the pressure variations in a sound wave to be adiabatic, whereas Laplace showed that they are isothermal.

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

Assertion (A): Beats are produced when two sound waves of nearly equal but slightly different frequencies are superposed, and the beat frequency equals the absolute difference of the two source frequencies.

Reason (R): The principle of superposition states that the displacement at any point in space is the algebraic sum of the displacements due to all the waves passing through that point.

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

1. I. In a transverse wave on a string, particles oscillate perpendicular to the direction of wave propagation.
2. II. Sound waves in air are transverse.
3. III. Electromagnetic waves are transverse and can travel through vacuum.

Examine the following statements about waves:

1. aOnly Statements I and III are correct
2. bAll three statements are correct
3. cOnly Statement III is correct
4. dOnly Statements I and II are correct

1. I. The wavelength of a sound wave depends on the medium.
2. II. The frequency of a sound wave is determined by its source and is independent of the medium.
3. III. The amplitude of a wave determines its loudness (for sound) or brightness (for light).

Examine the following statements about wave properties:

1. aOnly Statements I and III are correct
2. bAll three statements are correct
3. cOnly Statement III is correct
4. dOnly Statements I and II are correct

1. I. Standing waves transport no net energy along the string.
2. II. Nodes are points of zero displacement; antinodes are points of maximum displacement.
3. III. The distance between two successive nodes (or two successive antinodes) equals the wavelength.

Examine the following statements about standing waves:

1. aOnly Statement I is correct
2. bAll three statements are correct
3. cOnly Statements II and III are correct
4. dOnly Statements I and II are correct

1. I. A pipe closed at one end has only odd harmonics of its fundamental frequency.
2. II. A pipe open at both ends has all integer multiples of the fundamental.
3. III. A taut string of length \(L\) fixed at both ends has fundamental frequency \(v/(2L)\).

Examine the following statements about pipe and string harmonics:

1. aOnly Statements II and III are correct
2. bOnly Statements I and II are correct
3. cAll three statements are correct
4. dOnly Statement III is correct

Match each wave type in List I with a correct property in List II:

1. a(i)-S, (ii)-Q, (iii)-R, (iv)-P
2. b(i)-Q, (ii)-P, (iii)-R, (iv)-S
3. c(i)-P, (ii)-R, (iii)-Q, (iv)-S
4. d(i)-P, (ii)-Q, (iii)-R, (iv)-S

Match each wave in List I with its speed in List II:

1. a(i)-R, (ii)-P, (iii)-S, (iv)-Q
2. b(i)-R, (ii)-Q, (iii)-S, (iv)-P
3. c(i)-Q, (ii)-R, (iii)-P, (iv)-S
4. d(i)-R, (ii)-S, (iii)-Q, (iv)-P

Match each wave phenomenon in List I with its description in List II:

1. a(i)-P, (ii)-Q, (iii)-R, (iv)-S
2. b(i)-Q, (ii)-P, (iii)-R, (iv)-S
3. c(i)-R, (ii)-Q, (iii)-P, (iv)-S
4. d(i)-S, (ii)-Q, (iii)-R, (iv)-P

A wave on a string has wavefunction \(y = 0.01\sin(2x - 200t)\) (SI units). The maximum particle speed is:

1. a\(200\ \text{m/s}\)
2. b\(0.01\ \text{m/s}\)
3. c\(2\ \text{m/s}\)
4. d\(100\ \text{m/s}\)

A sonometer wire of length 1 m vibrates in its fundamental mode at 300 Hz under tension 100 N. If the wire is replaced with a thicker wire of the same material whose linear mass density is double, while length and tension remain unchanged, the new fundamental frequency is approximately:

1. a425 Hz
2. b150 Hz
3. c212 Hz
4. d600 Hz

A wave equation is \(y = 0.05\sin(4x - 80t)\) (SI units). The wavelength is:

1. a\(\pi\ \text{m}\)
2. b\(\dfrac{\pi}{2}\ \text{m}\)
3. c\(2\pi\ \text{m}\)
4. d\(0.05\ \text{m}\)

Based on the passage, the wave speed \(v\) on the string is:

1. a\(50\ \text{m/s}\)
2. b\(10\ \text{m/s}\)
3. c\(1000\ \text{m/s}\)
4. d\(100\ \text{m/s}\)

Based on the passage (\(y = 0.05\sin(20x - 2000t)\)), the wavelength is:

1. a\(\dfrac{\pi}{10}\ \text{m}\)
2. b\(\dfrac{\pi}{100}\ \text{m}\)
3. c\(10\pi\ \text{m}\)
4. d\(\pi\ \text{m}\)

Based on the passage, the frequency of the wave is:

1. a\(\dfrac{100}{\pi}\ \text{Hz}\)
2. b\(\dfrac{1000}{\pi}\ \text{Hz}\)
3. c\(\dfrac{10}{\pi}\ \text{Hz}\)
4. d\(2000\ \text{Hz}\)

Based on the passage, the wave speed computed from \(\omega/k\) matches the speed computed from \(\sqrt{T/\mu}\). The two values are:

1. aboth \(100\ \text{m/s}\), consistent
2. b\(100\ \text{m/s}\) and \(10\ \text{m/s}\), inconsistent
3. c\(10\ \text{m/s}\) and \(100\ \text{m/s}\), inconsistent
4. d\(1000\ \text{m/s}\) and \(100\ \text{m/s}\), inconsistent