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

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

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

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

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

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

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

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

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

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.

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

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.

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

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

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

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

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

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.

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

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

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

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

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

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

$f=\frac{1}{T}$ 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.