Take the first derivative of the position function with respect to time: $v(t) = \frac{dx(t)}{dt} = -A\omega \sin(\omega t + \phi)$.

Take the first derivative of the velocity function with respect to time: $a(t) = \frac{dv(t)}{dt} = -A\omega^2 \cos(\omega t + \phi)$.

Use the formula: $T = 2\pi \sqrt{\frac{m}{k}}$, where m is the mass and k is the spring constant.

Use the formula: $T = 2\pi \sqrt{\frac{L}{g}}$, where L is the length of the pendulum and g is the acceleration due to gravity.

Sum the kinetic and potential energies: $ME = K + U$, where $K = \frac{1}{2}mv^2$ and $U = \frac{1}{2}kx^2$ (for a spring) or $U = mgh$ (for gravity).

Mass-Spring: $T = 2\pi \sqrt{\frac{m}{k}}$ (depends on mass and spring constant). Pendulum: $T = 2\pi \sqrt{\frac{L}{g}}$ (depends on length and gravity).

Kinetic: Max at equilibrium, zero at max displacement. Potential: Zero at equilibrium, max at max displacement. Total energy is conserved.

Periodic motion where the restoring force is proportional to the displacement from equilibrium.

The force that always directs toward the equilibrium position and is proportional to the displacement ($F = -kx$).

The maximum displacement of the object from its equilibrium position.

A measure of how rapidly the oscillations occur, related to the period (T) by $\omega = \frac{2\pi}{T}$.

The time it takes for one complete cycle of the motion.

It represents the initial position of the oscillating object at time t=0.