Representing and Analyzing SHM - AP Physics 1

$v_{max} = 1$ m/s. Maximum speed occurs at equilibrium position.

At maximum displacement. Restoring force and acceleration are greatest at amplitude.

A swinging pendulum. Pendulums exhibit simple harmonic motion for small angles.

$f = 4$ Hz. Use $f = \frac{1}{T}$ to convert period to frequency.

$\omega = 2\pi f$. Angular frequency is $2\pi$ times the regular frequency.

$T \approx 2.01$ s. Use pendulum formula with given values.

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

$T = 0.2$ s. Use $T = \frac{1}{f}$ to find period from frequency.

A sine or cosine wave. Displacement varies sinusoidally with time in SHM.

$\frac{\pi}{2}$ radians. Velocity leads displacement by 90 degrees in phase.

$v = v_{max}$. At equilibrium, all energy is kinetic, reaching maximum speed.

The maximum displacement from equilibrium. Amplitude represents the extent or range of oscillation.

Motion where acceleration is proportional to displacement and directed towards equilibrium. Restoring force follows $F = -kx$, creating sinusoidal motion.

Damping reduces amplitude over time. Energy loss causes oscillation amplitude to decrease gradually.

$\omega = 4\pi$ rad/s. Use $\omega = \frac{2\pi}{T}$ to convert period to angular frequency.

$f = \frac{1}{T}$. Frequency and period are reciprocals of each other.

$\pi$ radians out of phase. Acceleration and velocity are opposite in phase.

$x(1) = 0$. Use $x(t) = A\cos(\omega t)$ with given values.

Kinetic and potential energy. Energy continuously converts between these two forms during oscillation.

Amplitude does not affect the period. Period depends only on system parameters, not initial conditions.

Energy is proportional to the square of amplitude. From $E = \frac{1}{2}kA^2$, energy increases with $A^2$.

Longer length increases period. Period increases with square root of length.

$x(0) = A$. At $t=0$ with $\phi=0$, $\cos(0) = 1$, so displacement equals amplitude.

Mass does not affect the period. Pendulum period depends only on length and gravity.

$E = \frac{1}{2}kA^2$. Total energy equals maximum potential energy at amplitude.

When driving frequency matches natural frequency, maximizing amplitude. External driving at natural frequency causes large amplitude oscillations.

Zero. At maximum displacement, all energy is potential, none kinetic.

$v(t) = -A\omega\sin(\omega t + \phi)$. Velocity is the time derivative of displacement function.

A sine wave. Velocity varies sinusoidally, 90 degrees ahead of displacement.