Representing and Analyzing SHM - AP Physics C: Mechanics

$f = \frac{1}{T}$. Frequency counts cycles per unit time, period is time per cycle.

$x(0) = \sqrt{2} \text{ m}$. At $t = 0$: $x(0) = 2\cos(\frac{\pi}{4}) = 2 \times \frac{1}{\sqrt{2}} = \sqrt{2}$ m.

radians per second (rad/s). Angular frequency measures how fast the phase angle changes with time.

$\omega = \omega_0$. Resonance occurs when driving frequency equals natural frequency.

$K = \frac{1}{2}mv^2$. Standard kinetic energy formula for any moving object.

[T^-1]. Angular frequency has dimension of inverse time.

$T = 2\pi \sqrt{\frac{m}{k}}$. Derived by substituting $\omega = \sqrt{\frac{k}{m}}$ into $T = \frac{2\pi}{\omega}$.

$T = \frac{2\pi}{\omega}$. Period is the time for one complete oscillation cycle.

$U = \frac{1}{2}kx^2$. Elastic potential energy stored in the compressed or stretched spring.

$x(t) = A , \cos(\omega t + \phi)$. General form where $A$ is amplitude, $\omega$ is angular frequency, and $\phi$ is phase constant.

$\omega = \sqrt{\frac{k}{m}}$. Derived from $F = -kx$ and Newton's second law for oscillatory motion.

Motion where restoring force is proportional to displacement. Defining characteristic: $F = -kx$ creates oscillatory motion.

$E \propto A^2$. Total energy is proportional to amplitude squared.

Zero. At amplitude, velocity is zero so kinetic energy is zero.

$U = \frac{1}{2}kx^2$. Potential energy varies with displacement squared.

$E = 1 \text{ J}$. $E = \frac{1}{2}kA^2 = \frac{1}{2} \times 50 \times (0.2)^2 = 1$ J.

[T^-1]. Angular frequency has dimension of inverse time.

Equilibrium position. Zero displacement corresponds to passing through equilibrium.

$f = \frac{10}{2\pi} \text{ Hz}$. Frequency relates to angular frequency by $f = \frac{\omega}{2\pi}$.

Period increases with mass as $T = 2\pi\sqrt{\frac{m}{k}}$. Period increases with square root of mass.

Maximum velocity: $v_{max} = A\omega$. At equilibrium, kinetic energy is maximum and equals $A\omega$.

$\omega = \sqrt{\frac{k}{m}}$. Fundamental relationship for harmonic oscillator systems.

$T = 2\pi \sqrt{\frac{1}{9.8}} \text{ s}$. $T = 2\pi\sqrt{\frac{1}{9.8}} = 2\pi\sqrt{0.102} \approx 2.0$ s.

Length of pendulum and gravitational acceleration. Period is independent of mass and amplitude for small oscillations.

Amplitude. Amplitude is the maximum distance from equilibrium position.

$\omega = 5 \text{ rad/s}$. $\omega = \sqrt{\frac{k}{m}} = \sqrt{\frac{100}{4}} = \sqrt{25} = 5$ rad/s.

Amplitude does not affect the period. Period depends only on system parameters $m$ and $k$, not amplitude.

$x(t) = A \cos(\omega t + \phi)$. Standard position function for simple harmonic motion.

Equal to total mechanical energy: $E = \frac{1}{2}kA^2$. At maximum displacement, all energy is potential energy.