MCAT Chemical and Physical Foundations of Biological Systems Flashcards: 4a Periodic Motion Mechanical Waves

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This deck focuses on 4a Periodic Motion Mechanical Waves, giving you a quick way to review the definitions, rules, and examples that matter most for MCAT Chemical and Physical Foundations of Biological Systems.

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Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is the maximum acceleration of a mass in SHM in terms of $A$ and $\omega$?

Answer: $a_{\max}=A\omega^2$. Maximum acceleration in SHM happens at maximum displacement, equaling amplitude times angular frequency squared from the defining equation.

Flashcard 2: Find the new period of a simple pendulum if its length changes from $L$ to $4L$ (same $g$).

Answer: $T\rightarrow 2T$. Pendulum period depends on square root of length, doubling when length quadruples as $T\propto\sqrt{L}$ under constant gravity.

Flashcard 3: Find the new period of a mass-spring system if the spring constant changes from $k$ to $4k$ (same $m$).

Answer: $T\rightarrow \frac{T}{2}$. Period inversely scales with square root of spring constant, halving when constant quadruples per $T\propto^1/\sqrt{k}$.

Flashcard 4: Find the new period of a mass-spring system if the mass changes from $m$ to $4m$ (same $k$).

Answer: $T\rightarrow 2T$. Period scales with square root of mass, so quadrupling mass doubles the period via $T\propto\sqrt{m}$ in mass-spring systems.

Flashcard 5: Find the period $T$ if a wave has frequency $f=8\ \text{Hz}$.

Answer: $T=\frac{1}{8}\ \text{s}=0.125\ \text{s}$. Period is the reciprocal of frequency, converting cycles per second to time per cycle for oscillatory or wave motion.

Flashcard 6: Find the frequency if a wave has speed $v=20\ \text{m/s}$ and wavelength $\lambda=5\ \text{m}$.

Answer: $f=4\ \text{Hz}$. Frequency derives from speed over wavelength, using $f=v/\lambda$ for the provided wave parameters.

Flashcard 7: Find the wavelength if a wave has speed $v=12\ \text{m/s}$ and frequency $f=3\ \text{Hz}$.

Answer: $\lambda=4\ \text{m}$. Wavelength calculates as speed divided by frequency, applying the fundamental wave relation $v=f\lambda$ to given values.

Flashcard 8: Identify the phase difference in radians for two points separated by $\frac{\lambda}{4}$ on a sinusoidal wave.

Answer: $\Delta\phi=\frac{\pi}{2}$. For sinusoidal waves, phase shifts by $2\pi$ per wavelength, so quarter-wavelength gives $\pi/2$ radians difference.

Flashcard 9: Identify the phase difference in radians for two points separated by $\frac{\lambda}{2}$ on a sinusoidal wave.

Answer: $\Delta\phi=\pi$. Phase difference scales with path length over wavelength times $2\pi$, yielding $\pi$ for half-wavelength separation in sinusoidal waves.

Flashcard 10: What physical property of the medium determines wave speed, and what property does not determine it?

Answer: Speed set by medium; frequency set by source (not medium). Medium properties like density and elasticity dictate wave speed, while source determines frequency independently of the medium.

Flashcard 11: What is the key distinction between transverse and longitudinal mechanical waves?

Answer: Transverse: oscillation $\perp$ travel; longitudinal: $\parallel$ travel. Wave types differ by particle oscillation direction relative to propagation: perpendicular for transverse, parallel for longitudinal.

Flashcard 12: What is the definition of wavelength $\lambda$ for a traveling periodic wave?

Answer: Distance between points in phase (e.g., crest to crest). Wavelength measures the spatial period of a wave, defined as the shortest distance between identical phase points in the cycle.

Flashcard 13: State the wave speed on an ideal string in terms of tension $T$ and linear density $\mu$.

Answer: $v=\sqrt{\frac{T}{\mu}}$. String wave speed balances tension restoring force and inertial mass density, yielding the square root expression for transverse waves.

Flashcard 14: State the speed of a wave in terms of frequency $f$ and wavelength $\lambda$.

Answer: $v=f\lambda$. Wave speed equals frequency times wavelength, relating temporal and spatial periodicity for propagating waves.

Flashcard 15: What is the kinetic energy of a mass in SHM in terms of $m$ and instantaneous speed $v$?

Answer: $K=\frac{1}{2}mv^2$. Kinetic energy in SHM represents motion energy, calculated classically as half mass times velocity squared at any instant.

Flashcard 16: What is the spring potential energy as a function of displacement $x$ for an ideal spring?

Answer: $U=\frac{1}{2}kx^2$. Spring potential stores elastic energy quadratically with displacement, following Hooke's law integration for conservative force.

Flashcard 17: State the total mechanical energy of an ideal mass-spring oscillator in terms of $k$ and amplitude $A$.

Answer: $E=\frac{1}{2}kA^2$. Total energy conserves as maximum potential energy at amplitude extremes, expressed via spring constant and squared amplitude.

Flashcard 18: What condition must hold for a pendulum to be treated as simple harmonic motion?

Answer: Small angles: $\sin\theta\approx\theta$ (in radians). The small-angle approximation linearizes the pendulum equation, enabling SHM by equating sine to the angle in radians.

Flashcard 19: State the period of a small-angle simple pendulum in terms of length $L$ and gravity $g$.

Answer: $T=2\pi\sqrt{\frac{L}{g}}$. For small angles, pendulum period approximates from torque balance, scaling with square root of length over gravitational acceleration.

Flashcard 20: State the angular frequency of a mass-spring oscillator in terms of $m$ and $k$.

Answer: $\omega=\sqrt{\frac{k}{m}}$. Angular frequency emerges from the SHM equation of motion, as the square root of spring constant over mass for oscillatory behavior.

Flashcard 21: State the period of a mass-spring oscillator in terms of mass $m$ and spring constant $k$.

Answer: $T=2\pi\sqrt{\frac{m}{k}}$. The period derives from solving the differential equation for a mass-spring system, depending on the square root of mass over spring constant.

Flashcard 22: State the SHM relation between acceleration and displacement for a mass on a spring.

Answer: $a=-\omega^2 x$. In SHM, acceleration is proportional to negative displacement, with $\omega^2$ as the constant from restoring force dynamics.

Flashcard 23: What is the maximum speed of a mass in SHM in terms of $A$ and $\omega$?

Answer: $v_{\max}=A\omega$. Maximum speed in SHM occurs at equilibrium, derived from velocity function or energy conservation as amplitude times angular frequency.

Flashcard 24: State the standard displacement function for simple harmonic motion using amplitude $A$ and phase $\phi$.

Answer: $x(t)=A\cos(\omega t+\phi)$. This equation models oscillatory displacement as a cosine function, capturing amplitude, angular frequency, time, and initial phase.

Flashcard 25: What is the relationship among period $T$, frequency $f$, and angular frequency $\omega$?

Answer: $f=\frac{1}{T}$ and $\omega=2\pi f=\frac{2\pi}{T}$. Frequency is the reciprocal of period, and angular frequency equals $2\pi$ times frequency, linking time-based oscillatory parameters.