Periodic Motion and Mechanical Waves (4A)
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MCAT Chemical and Physical Foundations of Biological Systems › Periodic Motion and Mechanical Waves (4A)
A small object undergoes uniform circular motion at constant speed, used as an analogy for periodic biological rhythms. Which statement best reflects the relationship between angular speed $\omega$ and period $T$?
(Assume one complete revolution per period.)
$\omega$ increases when $T$ increases because angular speed and period are directly proportional.
$\omega$ is measured in meters per second, so it does not depend on $T$.
$\omega = 2\pi/T$, so shorter periods correspond to larger angular speeds.
$\omega = T/2\pi$, so longer periods correspond to larger angular speeds.
Explanation
This question tests angular speed in periodic motion. Angular speed ω=2π/T, so ω inversely proportional to T, larger for shorter periods. In this circular motion analogy for rhythms, the relation holds. The correct answer B follows because shorter T means higher ω. Distractor C reverses the formula, a calculation error. In periodic motion, use ω=2πf=2π/T. Distinguish ω (rad/s) from linear speed v=rω.
A student measures wave speed in a fluid-filled tube (representing sound propagation in airway mucus) by sending a sinusoidal pressure wave down the tube. The source frequency is doubled while the fluid properties and tube remain unchanged. Which change is most consistent with wave behavior in this setup?
(Assume wave speed depends only on the medium.)
The wave amplitude must decrease because higher frequency waves carry less energy.
The wave speed doubles because frequency determines wave speed in any medium.
The wavelength is halved because wave speed is constant and $v = f\lambda$.
The wavelength doubles because frequency and wavelength increase together in a fixed medium.
Explanation
This question tests mechanical wave properties, specifically the relationship between frequency, wavelength, and speed. The wave speed in a medium is constant and determined by the medium's properties, with v = fλ holding for sinusoidal waves. In this fluid-filled tube modeling airway mucus, doubling the frequency while keeping the medium unchanged maintains constant wave speed. The correct answer B follows because halving the wavelength satisfies v = fλ when f doubles and v is fixed. Distractor A fails due to the misconception that wavelength increases with frequency, but they are inversely related at constant speed. To verify in similar questions, calculate wavelength changes using λ = v/f. Always confirm if the medium's properties are altered, as that affects v.
A tuning-fork source drives pressure waves in air for a hearing study. The intensity at a point is increased without changing frequency. Which change is most consistent with wave mechanics in a linear medium?
(Assume no attenuation changes.)
The wavelength increases because intensity and wavelength are directly proportional.
The frequency decreases because energy conservation requires fewer oscillations per second.
The wave speed increases because higher intensity means faster propagation.
The amplitude of the pressure oscillations increases.
Explanation
This question tests mechanical wave intensity and properties. Intensity I is proportional to amplitude squared for waves, with frequency fixed here. In this hearing study, increasing intensity at constant frequency raises pressure amplitude. The correct answer A follows because higher I means larger oscillations, as I ∝ $(amplitude)^2$ in linear media. Distractor B assumes speed changes with intensity, a misconception for linear waves where v is medium-dependent. In intensity problems, relate changes to amplitude via I ∝ $A^2$ $f^2$. Verify if the medium remains linear, as nonlinearity could alter behavior.
In an ultrasound phantom, a transducer emits a pulse that reflects from a boundary and returns to the detector. The medium’s wave speed is unchanged, but the pulse frequency is increased to improve resolution. Which statement is most consistent with the physics of wave propagation in the same medium?
(Assume linear acoustics.)
The pulse travels faster because higher frequency implies higher speed in the same medium.
The wavelength decreases, which can improve spatial resolution.
The boundary reflection disappears because higher frequency waves cannot reflect.
The return time decreases because frequency determines time-of-flight.
Explanation
This question tests mechanical wave propagation and resolution in ultrasound. Higher frequency reduces wavelength via λ = v/f, improving resolution as smaller wavelengths distinguish closer features. In this ultrasound phantom, increasing frequency with constant v shortens λ for better spatial resolution. The correct answer B follows because shorter wavelength directly enhances resolution without altering speed or reflection. Distractor A assumes speed increases with frequency, a misconception ignoring that v depends on the medium. For other wave resolution problems, recall resolution improves with shorter λ. Check if frequency changes affect attenuation, though not relevant here.
A pulse travels along a taut string used as a simplified nerve-axon analog for mechanical signaling. The string tension is increased while the linear mass density is unchanged. Which change is most consistent with the wave speed on the string?
(Use $v = \sqrt{T/\mu}$ conceptually.)
Wave speed is unchanged because only frequency determines speed.
Wave speed becomes zero because a taut string cannot support transverse waves.
Wave speed increases because higher tension increases restoring force per displacement.
Wave speed decreases because higher tension reduces transverse displacement.
Explanation
This question tests wave speed on strings. Wave speed v = √(T/μ), so increasing T increases v for fixed μ. In this nerve-axon analog, higher tension enhances restoring force, speeding the wave. The correct answer A follows because greater T directly raises v per the formula. Distractor B assumes tension reduces displacement, but it actually accelerates propagation. In string wave problems, use v ratios like v_new/v_old = √(T_new/T_old). Ensure μ is constant, as changes would inversely affect v.
In a resonance-tube demonstration relevant to speech acoustics, a tube is closed at one end and open at the other. The first loud resonance occurs at tube length $L$. Which statement is most consistent with the standing-wave pattern at this first resonance?
(Assume speed of sound is constant.)
A displacement node occurs at the open end and a displacement antinode at the closed end.
No standing wave forms at first resonance because only higher harmonics are allowed.
A displacement node occurs at the closed end and a displacement antinode at the open end.
Displacement antinodes occur at both ends because air is free to move everywhere.
Explanation
This question tests standing waves in open-closed tubes. For the fundamental resonance, a displacement node is at the closed end and antinode at the open end, with L = λ/4. In this speech acoustics demo, first resonance at L fits this pattern. The correct answer C follows because it correctly identifies node and antinode positions. Distractor A reverses them, a misconception confusing pressure and displacement. In tube resonance questions, recall odd harmonics for open-closed tubes. Use L = (2n-1)λ/4 for nth harmonic verification.
Two speakers emit coherent sound waves of the same frequency toward a point in space, modeling interference in an audiology setup. At the point, destructive interference is observed. Which condition is most consistent with this outcome?
(Assume equal amplitudes.)
The waves have different speeds, so they cancel regardless of phase.
The path length difference is a half-integer multiple of $\lambda$.
The path length difference is an integer multiple of $\lambda$.
The waves must have different frequencies to cancel at a point.
Explanation
This question tests wave interference principles. Destructive interference for equal-amplitude coherent waves occurs when path difference δ = (m + 1/2)λ, a half-integer multiple. In this audiology setup, the condition for cancellation matches this. The correct answer B follows because destructive interference requires odd multiples of λ/2 path difference. Distractor A describes constructive interference, a common confusion of conditions. For interference problems, use δ = mλ for constructive, (m+1/2)λ for destructive. Assume coherence unless stated otherwise.
A simple harmonic oscillator undergoes motion $x(t)=A\cos(\omega t)$. At $t=0$, the oscillator is at maximum displacement. Which statement is most consistent with the velocity at $t=0$?
(Assume ideal SHM.)
Velocity is undefined because cosine motion has discontinuities at turning points.
Velocity is maximal because displacement is maximal.
Velocity is constant because acceleration is zero at maximum displacement.
Velocity is zero because the oscillator reverses direction at maximum displacement.
Explanation
This question tests simple harmonic motion kinematics. In x(t)=A cos(ωt), velocity v(t)=-Aω sin(ωt), so at t=0 (max x), v=0. At maximum displacement, the oscillator momentarily stops before reversing. The correct answer B follows because turning points have zero velocity. Distractor A confuses max displacement with max velocity, which occurs at equilibrium. In SHM, use v=0 at extrema, max at x=0. Derive from energy: KE=0 when PE max.
A student compares longitudinal and transverse mechanical waves in a lab relevant to respiratory acoustics. Which statement is most consistent with the distinction between these wave types?
(Assume both propagate in a material medium.)
In longitudinal waves, particle oscillations are parallel to the direction of propagation.
Longitudinal waves cannot carry energy because they do not have crests and troughs.
Transverse waves require a vacuum, whereas longitudinal waves require a medium.
In transverse waves, particle oscillations are parallel to the direction of propagation.
Explanation
This question tests classification of mechanical waves. Longitudinal waves have particle motion parallel to propagation, transverse perpendicular. In respiratory acoustics, sound is longitudinal with parallel oscillations. The correct answer A follows because it accurately distinguishes the types. Distractor B reverses the definitions, a common mix-up. For wave type questions, recall direction of oscillation relative to propagation. Note both require media, unlike electromagnetic waves.
In a cochlea-inspired model, a traveling wave on a membrane shows maximal displacement at a location where the local resonant frequency matches the stimulus frequency. If the stimulus frequency increases, which shift is most consistent with this resonance-based mapping?
(Assume different membrane regions have different natural frequencies.)
The location of maximal displacement shifts to a region with higher natural frequency.
The maximal displacement disappears because resonance cannot occur at higher frequency.
The location of maximal displacement remains fixed because frequency only affects amplitude.
The maximal displacement shifts to a region with lower natural frequency because higher frequency waves travel farther.
Explanation
This question tests resonance in wave systems like the cochlea. Maximal displacement occurs where local natural frequency matches stimulus frequency. Increasing stimulus frequency shifts peak to higher-frequency regions. The correct answer A follows because resonance mapping places higher f at corresponding sites. Distractor B ignores frequency dependence, assuming fixed location. In tonotopic models, map frequency to position via resonance. Check if system has graded properties affecting local frequencies.