Revise Simple harmonic oscillations for Cambridge 9702 (17.1)

17.1 Simple harmonic oscillations.

An oscillation is defined as repeated back and forth movements on either side of any equilibrium position.

When the object stops oscillating it returns to its equilibrium position.

An oscillation is a more specific term for a vibration.

An oscillator is a device that works on the principles of oscillations.

Oscillating systems can be represented by displacement-time graphics.

Displacement (x): Instantaneous distance from equilibrium.

Amplitude ($x_0$ or A): Maximum displacement from equilibrium.

Period (T): Time for one complete oscillation cycle (in seconds).

Frequency (f): Number of cycles per second ($f = 1/T$, in Hertz).

Angular Frequency ($\omega$): Rate of change of phase angle ($\omega = 2\pi f = 2\pi/T$, in rad/s).

Velocity (v): Rate of change of displacement.

Maximum velocity ($v_0 = \omega x_0$) occurs at equilibrium ($x=0$).

Velocity can be found using $v = \pm \omega \sqrt{x_0^2 - x^2}$.

Acceleration (a): Rate of change of velocity, $a = -\omega^2 x$.

Displacement and Acceleration: $180^\circ$ ($\pi$ radians) out of phase. When x is max positive, a is max negative.

Displacement and Velocity: $90^\circ$ ($\pi/2$ radians) out of phase. When x is max, v is zero; when x is zero, v is max.

In Phase: Two oscillations move together, peaking and troughing at the same time (phase difference = $2\pi n$).

Completely Out of Phase: Two oscillations move exactly opposite (phase difference = $(2n+1)\pi$).

Displacement-time (x-t) graph: A sine or cosine curve. The peak value is the amplitude ($x_0$), and the time for one full wave is the period (T).

Velocity-time (v-t) graph: Also a sinusoidal curve, representing the gradient of the x-t graph. It leads the displacement by $90^\circ$ ($\pi/2$ radians). When displacement is zero, velocity is maximum or minimum.

Acceleration-time (a-t) graph: Another sinusoidal curve, representing the gradient of the v-t graph. It is $180^\circ$ ($\pi$ radians) out of phase with displacement, meaning it's an inverted version of the x-t graph (scaled by $\omega^2$).

Energy-displacement graph: A graph of energy vs. displacement shows a constant total energy line, a parabola for potential energy ($E_P \propto x^2$), and an inverted parabola for kinetic energy.

Kinetic Energy (KE): Energy of motion, maximal at the equilibrium position ($x=0$). Given by $E_K = \frac{1}{2} m \omega^2 (x_0^2 - x^2)$.

Potential Energy (PE): Stored energy, maximal at the amplitude positions ($x=\pm x_0$). Given by $E_P = \frac{1}{2} m \omega^2 x^2$.

Energy Conservation: As the oscillator moves, energy continuously transforms between KE and PE, but their sum, the total energy, remains constant.

Total Energy ($E_{total}$): The sum $E_K + E_P$ is always constant and equals the maximum potential or kinetic energy: $E_{total} = \frac{1}{2} m \omega^2 x_0^2$.

Free Oscillations: Oscillations without any external driving force, at a system's natural frequency.

Underdamping (Light Damping): Amplitude slowly decays over many cycles, like a swinging pendulum.

Critical Damping: System returns to equilibrium in the shortest time possible without oscillating. Ideal for car suspensions.

Overdamping (Heavy Damping): Damping is too strong, causing a slow return to equilibrium than critical damping, also without oscillating.