Oscillations and Waves Questions and Answers

Answer: A

The speed of sound is proportional to \(\sqrt{T}\).

Laplace's formula gives \(v = \sqrt{\frac{\gamma P}{\rho}}\). For an ideal gas \(\frac{P}{\rho} = \frac{RT}{M}\), so

\[ v = \sqrt{\frac{\gamma RT}{M}} \propto \sqrt{T} \]

At a given temperature, changing the pressure changes the density in the same proportion, so the speed does not depend on density on its own. Speed increases with temperature, but as its square root, not as T or \(T^{2}\). \(\sqrt{\rho}\) has the density on the wrong side; in the formula it appears in the denominator.

Answer: E

For small oscillations the period depends on its length.

For small angular displacements the motion is simple harmonic with period

\[ T = 2\pi\sqrt{\frac{l}{g}} \]

It depends only on the length l and the acceleration due to gravity g. It does not depend on the amplitude (this is the isochronism of small oscillations), on the mass of the bob, on the energy of the swing, or on the phase constant, which only fixes where in its cycle the pendulum starts.

Answer: D

The resultant amplitude is 0.

Two waves of amplitudes \(A_{1}\) and \(A_{2}\) with phase difference \(\phi\) combine to an amplitude

\[ R = \sqrt{A_{1}^{2} + A_{2}^{2} + 2A_{1}A_{2}\cos\phi} \]

Opposite phase means \(\phi = \pi\), so \(\cos\phi = -1\) and \(R = |A_{1} - A_{2}| = 0\) for equal amplitudes: complete destructive interference. 2A is the in-phase (constructive) result, and \(A^{2}\) confuses amplitude with intensity.

Answer: A

The ratio is 2 : 1.

An open pipe has antinodes at both ends. Its modes fit whole half-wavelengths into the length L: \(L = \frac{n\lambda_{n}}{2}\), so \(\lambda_{n} = \frac{2L}{n}\).

\[ \lambda_{1} : \lambda_{2} = 2L : L = 2 : 1 \]

The frequencies are in the reverse ratio, 1 : 2, the trap answer. 1 : 3 describes the frequencies of the first two modes of a closed pipe, which has only odd harmonics.

Answer: A

The work done is \(\frac{1}{2}K(x_{2}^{2} - x_{1}^{2})\).

The applied force needed to hold the spring at extension \(x\) is \(Kx\). The work done in going from \(x_{1}\) to \(x_{2}\) is:

\[ \begin{aligned} W &= \int_{x_{1}}^{x_{2}}Kx\,dx \\ &= \frac{1}{2}K(x_{2}^{2} - x_{1}^{2}) \end{aligned} \]

This is the increase in stored elastic energy \(\frac{1}{2}Kx^{2}\). The option with \(\frac{1}{4}K(x_{1}^{2} - x_{2}^{2})\) has the wrong factor and the wrong sign. The products \(x_{1}x_{2}\) and \(x_{1}^{2}x_{2}^{2}\) do not come from integrating \(Kx\).

Answer: E

The correct statement is at O, \(a\) = 0.

In SHM, \(a = -\omega^{2}x\): the acceleration is proportional to the displacement from the mean position. At \(O\), \(x = 0\), so \(a = 0\), while the speed is greatest there.

At the extreme positions \(A\) and \(B\), \(|x|\) is greatest, so the acceleration is maximum, not zero, and the velocity is zero. 'At \(O\), \(a\) is maximum' reverses this, and 'at \(O\), \(a\) and \(v\) are maximum' is half right: \(v\) is maximum there but \(a\) is zero.

Answer: B

The displacement after 3 s is 5 m.

Substitute \(t = 3\) s:

\[ \begin{aligned} x(3) &= 5\cos(4\pi \times 3) = 5\cos 12\pi \\ &= 5 \times 1 = 5\ \text{m} \end{aligned} \]

The angular frequency \(4\pi\) rad/s gives a period \(T = \frac{2\pi}{4\pi} = 0.5\) s, so 3 s is exactly 6 whole oscillations and the particle is back at its starting point, the positive extreme \(x = 5\) m.

10 m is twice the amplitude, a distance the particle never reaches from the mean position. 2, 3 and 4 m would need a phase that is not a whole number of cycles.

Answer: C

The frequency difference is 8 Hz.

The beat frequency equals the difference between the two source frequencies:

\[ \begin{aligned} |f_{1} - f_{2}| &= \frac{24\ \text{beats}}{3\ \text{s}} \\ &= 8\ \text{Hz} \end{aligned} \]

12, 4 and 2 divide 24 by the wrong number, and 3 is the time interval itself. A beat is one cycle of loudness, so no factor of 2 enters.

Answer: B

A wavefront is a surface of constant phase.

In Huygens' construction, a wavefront is the locus of all points vibrating in the same phase. Each point on it acts as a source of secondary wavelets, and the envelope of those wavelets gives the next wavefront. Rays are normal to the wavefronts.

A single ray is a line, the direction of propagation, not a surface. Points on a wavefront are by definition in phase, so 'varying phase' contradicts it. A region where crests and troughs overlap describes interference, not a wavefront.

Answer: C

The false statement is The spring constant is independent of the elastic properties of the spring.

The spring constant is a measure of the spring's stiffness, and it depends directly on the elastic properties (the material's modulus) and on the spring's dimensions. So this statement is false.

The others are true: stiff springs have large k and soft springs small k; for small oscillations the restoring force is proportional to displacement, giving SHM; and the period \(T = 2\pi\sqrt{\frac{m}{k}}\) depends on k.