Mechanical Waves and Sound: questions with solutions
3 free questions from this chapter (the full set has 30). Marking: Multi-correct: +4 all correct, +1 per correct option if none wrong, −2 if any wrong · Numerical: +4 / 0 · Matching: +3 / −1.
Question 1Level 2 · ExamSingle correct
A sinusoidal transverse wave travels along a long taut string lying on the x-axis, in the +x direction. Its amplitude is 4mm, wavelength 60cm and frequency 50Hz; displacements are along y. At t=0 the particle at x=0 has displacement y=+2mm and is moving in the −y direction. At t=0, the particle at x=+20cm has ______. Exactly one option is correct.
Ay=+2mm, moving in the −y direction with speed 1.09m/s
By=−2mm, moving in the −y direction with speed 1.09m/s
Cy=+2mm, moving in the +y direction with speed 1.09m/s
Dy=+4mm, momentarily at rest
Show answer and solution
Answer: C
For a wave moving along +x: y=Asin(kx−ωt+ϕ), A=4mm, k=2π/0.6m−1, ω=100πs−1.
At x=0, t=0: Asinϕ=A/2 and vy=∂y/∂t=−Aωcosϕ<0, so cosϕ>0 and ϕ=π/6.
At x=0.20m: kx=2π/3, phase =2π/3+π/6=5π/6.
y=Asin(5π/6)=+2mm; vy=−Aωcos(5π/6)=+Aω23=0.004×100π×0.866=1.09m/s, i.e. along +y.
Trap: a sign slip in vy=−Aωcos(kx−ωt+ϕ) gives option (A); forgetting the direction condition at x=0 (ϕ=5π/6) puts the particle at a trough instead.
Difficulty: Hard
Question 2Level 3 · AdvancedSingle correct
An ideal-gas mixture contains 1 mol of helium (monatomic) and 1 mol of oxygen (diatomic, rigid: translational and rotational modes only) at absolute temperature T. Molar masses are 4g/mol (He) and 32g/mol (O2). Sound propagation is adiabatic. The ratio of the speed of sound in the mixture to the speed of sound in pure helium at the same temperature is ______. Exactly one option is correct.
A1/5
B46/15
C14/75
D2/3
Show answer and solution
Answer: A
v=γRT/M with γ and M of the mixture.
Mixture: CV=21⋅23R+1⋅25R=2R, CP=3R, so γmix=3/2; Mmix=(4+32)/2=18g/mol.
Helium: γ=5/3, M=4g/mol.
(vHevmix)2=(5/3)/4(3/2)/18=5/121/12=51, so the ratio is 1/5≈0.447.
Trap: averaging γ arithmetically (γ=23/15) gives 46/15; using γ=7/5 gives 14/75; keeping γ=5/3 gives 2/3. The correct γ comes from the mole-weighted heat capacities.
Difficulty: Very hard
Question 3Level 2 · ExamSingle correct
A sonometer wire vibrating in its fundamental mode produces 4 beats per second with a tuning fork. When the tension in the wire is increased by 21% (vibrating length and wire unchanged), the wire, again in its fundamental mode, still produces 4 beats per second with the same fork. The frequency of the fork is ______. Exactly one option is correct.
A76Hz
B80Hz
C88Hz
D84Hz
Show answer and solution
Answer: D
f∝T, so the new wire frequency is 1.21f1=1.1f1.
Same beat count both times with f increased means the wire moved from 4Hz below the fork to 4Hz above it: f1=n−4, 1.1f1=n+4.
So 0.1f1=8, f1=80Hz, n=84Hz. (Both below or both above is impossible since f changed.)
Difficulty: Hard
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