Periodic Motion and Mechanical Waves (4A)

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MCAT Chemical and Physical Foundations of Biological Systems › Periodic Motion and Mechanical Waves (4A)

Questions 1 - 10
1

A researcher compares two sinusoidal waves traveling in the same direction through the same fluid: Wave 1 and Wave 2 have the same frequency, but Wave 2 has a larger amplitude. Which conclusion is most consistent with mechanical wave behavior in a linear medium?

Wave 2 travels faster because larger amplitude increases wave speed in the same medium

Wave 2 has a higher frequency because amplitude increases the number of cycles per second

Wave 2 has a longer wavelength because amplitude and wavelength are directly proportional

Wave 2 carries more energy, while wave speed and wavelength remain unchanged

Explanation

This question tests the independence of wave parameters in linear media. In a linear medium, wave speed depends only on medium properties (like density and elasticity), not on wave amplitude. Since both waves travel through the same fluid at the same frequency, they have identical wave speeds and wavelengths. The key difference is energy: wave energy is proportional to amplitude squared, so Wave 2 carries more energy while maintaining the same propagation characteristics. Choice A incorrectly links amplitude to wave speed, confusing energy content with propagation speed. Remember: in linear media, changing amplitude affects energy and intensity but not frequency, wavelength, or wave speed.

2

A damped spring–mass system is used to model soft-tissue vibration after an impulse. Compared with an otherwise identical system with less damping, the more heavily damped system is observed to have a smaller oscillation amplitude after the same initial displacement. Which statement is most consistent with periodic motion in damped systems?

Greater damping increases the system’s total mechanical energy, increasing amplitude over time

Greater damping increases the natural frequency, which necessarily increases amplitude

Greater damping converts oscillatory motion into constant-velocity linear motion with the same amplitude

Greater damping causes mechanical energy to be dissipated more rapidly, reducing amplitude over time

Explanation

This question tests the effect of damping on oscillatory motion. Damping represents energy dissipation mechanisms that convert mechanical energy into other forms (typically heat), reducing the system's total mechanical energy over time. Greater damping means energy is lost more rapidly, resulting in smaller oscillation amplitudes for the same initial conditions. The natural frequency may slightly decrease with heavy damping, but this doesn't increase amplitude. Choice B incorrectly claims damping adds energy to the system, contradicting the fundamental nature of dissipative forces. Remember: damping always removes energy from oscillatory systems, reducing amplitude over time—it never adds energy or increases amplitude.

3

A string fixed at both ends is used to model vibration of a biological filament. The string is driven at a frequency that produces the second harmonic. Which description is most consistent with the standing-wave pattern for the second harmonic?

A uniform displacement along the entire string with no nodes

Two antinodes with a node at the center and nodes at both ends

One antinode at the center and nodes only at the ends

Three antinodes with two interior nodes and nodes at both ends

Explanation

This question tests standing wave patterns for strings fixed at both ends. The second harmonic (n=2) has a wavelength λ = L, where L is the string length, creating two half-wavelengths along the string. This produces two antinodes (maximum displacement points) with a node at the center, plus the required nodes at both fixed ends. The fundamental (n=1) would have only one antinode, while the third harmonic (n=3) would have three antinodes. Choice D describes the third harmonic pattern, not the second. A useful check: for the nth harmonic on a string fixed at both ends, there are n antinodes and (n+1) nodes total.

4

A Doppler ultrasound probe emits sound at frequency $f_0$ into tissue and receives echoes from blood moving directly toward the probe. Assume the speed of sound in tissue is approximately constant ($v = 1540\ \text{m/s}$) and the blood speed is much smaller than $v$. Which observation is most consistent with the Doppler effect for the received echo frequency compared with $f_0$?

The received frequency is higher than $f_0$ because motion toward the probe decreases the effective wavelength between wavefronts.

The received frequency is lower than $f_0$ because motion toward the probe increases wavelength.

The received frequency is higher than $f_0$ only if the emitted amplitude is increased.

The received frequency equals $f_0$ because sound speed in tissue is constant.

Explanation

This question tests the Doppler effect for sound waves when the source (blood) moves toward the observer (probe). When a source moves toward an observer, the wavefronts are compressed, decreasing the effective wavelength between successive crests. Since wave speed in the medium remains constant (v = 1540 m/s in tissue), and v = fλ, a decrease in wavelength must correspond to an increase in frequency. The received frequency is therefore higher than the emitted frequency f₀, confirming choice B. Choice A incorrectly states that motion toward the observer increases wavelength, which would occur if the source moved away. Choice C ignores the Doppler effect entirely. Choice D incorrectly links frequency shift to amplitude, when the Doppler shift depends only on relative motion. For Doppler problems, remember: source moving toward observer → higher frequency; source moving away → lower frequency. The shift magnitude depends on the ratio of source speed to wave speed.

5

Two identical loudspeakers emit the same single-frequency tone in phase and are positioned so that a listener’s ear is equidistant from both speakers. The listener then moves to a new point where the path length from Speaker 2 is longer by $\Delta L = \lambda/2$ (with $\lambda$ the wavelength in air). Assume equal amplitudes at the ear. Which outcome is most consistent with interference of the sound waves at the ear?

Greater loudness because longer path length increases wave amplitude at the ear.

Destructive interference because a half-wavelength path difference produces a $\pi$ phase shift.

No interference because sound waves do not superpose in air.

Constructive interference because a half-wavelength path difference produces in-phase arrival.

Explanation

This question tests wave interference based on path difference. When two coherent sources emit in phase, the phase difference at the observation point depends on the path difference. A path difference of ΔL = λ/2 corresponds to a phase difference of π radians (180°), since phase difference = (2π/λ) × path difference = (2π/λ) × (λ/2) = π. When two waves of equal amplitude arrive with opposite phase (π phase difference), they interfere destructively, producing zero net amplitude at that point. This confirms choice B is correct. Choice A incorrectly claims λ/2 path difference produces constructive interference, which would require integer multiples of λ. Choice C incorrectly denies that sound waves can interfere. Choice D incorrectly links path length to amplitude rather than phase. For interference problems, remember: path difference of nλ → constructive; path difference of (n + 1/2)λ → destructive, where n is an integer.

6

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 speed doubles because frequency determines wave speed in any medium.

The wavelength doubles because frequency and wavelength increase together in a fixed medium.

The wavelength is halved because wave speed is constant and $v = f\lambda$.

The wave amplitude must decrease because higher frequency waves carry less energy.

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.

7

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 return time decreases because frequency determines time-of-flight.

The boundary reflection disappears because higher frequency waves cannot reflect.

The wavelength decreases, which can improve spatial resolution.

The pulse travels faster because higher frequency implies higher speed in the same medium.

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.

8

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 path length difference is an integer multiple of $\lambda$.

The waves must have different frequencies to cancel at a point.

The waves have different speeds, so they cancel regardless of phase.

The path length difference is a half-integer multiple of $\lambda$.

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.

9

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 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.

The location of maximal displacement shifts to a region with higher natural frequency.

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.

10

A string fixed at both ends is driven at a frequency that produces a standing wave with three antinodes. Which statement is most consistent with the harmonic produced?

(Assume ideal string; length $L$.)

This is the fundamental, and the wavelength satisfies $\lambda = 2L$.

This is the third harmonic, and the wavelength satisfies $\lambda = 2L/3$.

This is not a harmonic because standing waves require only one antinode.

This is the second harmonic, and the wavelength satisfies $\lambda = L$.

Explanation

This question tests harmonics in standing waves. For fixed ends, nth harmonic has n antinodes, λ=2L/n; three antinodes is n=3, λ=2L/3. With three antinodes, it's the third harmonic. The correct answer A follows because it matches the harmonic and wavelength. Distractor B miscounts as second with wrong λ, confusing node count. In standing wave questions, count antinodes for n. Use f_n = n v /(2L) for frequency scaling.

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