Circular Motion and Simple Harmonic Motion

Flashcards reviewing key vocabulary, formulas, and concepts of circular motion, simple harmonic motion, resonance, and damping.

Centripetal force

A constant force applied always towards the centre of a circle, which is required to keep an object moving in a circle at constant speed.

Circular motion acceleration

An object moving in a circle at a constant speed is accelerating because its direction is always changing, which means velocity is always changing, where acceleration is defined as the change in velocity over time.

Angular speed ($\omega$)

The rate of rotation, calculated as $\omega = \frac{v}{r}$ or $\omega = 2\pi f$, where $\omega$ is in $\text{s}^{-1}$, $v$ is in $\text{m}\,\text{s}^{-1}$, $r$ is in $\text{m}$, and $f$ is in $\text{Hz}$.

Angular acceleration (in terms of angular velocity)

Acceleration expressed in terms of angular velocity as $a = \omega^2 r$, where $a$ is in $\text{m}\,\text{s}^{-2}$, $\omega$ is in $\text{s}^{-1}$, and $r$ is in $\text{m}$.

Angular acceleration (in terms of velocity)

Acceleration expressed in terms of linear velocity as $a = \frac{v^2}{r}$, where $a$ is in $\text{m}\,\text{s}^{-2}$, $v$ is in $\text{m}\,\text{s}^{-1}$, and $r$ is in $\text{m}$.

Centripetal force equations

Formulas used to determine centripetal force: $F = \frac{m v^2}{r}$ or $F = m \omega^2 r$, where $F$ is in $\text{N}$, $m$ is in $\text{kg}$, $v$ is in $\text{m}\,\text{s}^{-1}$, $\omega$ is in $\text{s}^{-1}$, and $r$ is in $\text{m}$.

Radian

The angle of a circle sector such that the radius is equal to the arc length, where $2\pi\text{ radians} = 360^\circ$.

Conditions for Simple Harmonic Motion (SHM)

Motion in which acceleration is proportional to displacement from the equilibrium point ($a \propto -x$) and acts towards the equilibrium point.

Displacement in SHM ($x$)

Displacement expressed as a trigonometric function of time ($t$) and angular speed ($\omega$): $x = A\cos(\omega t)$ or $x = A\sin(\omega t)$, where $x$ and $A$ are in $\text{m}$, $\omega$ is in $\text{s}^{-1}$, and $t$ is in $\text{s}$.

Maximum speed in SHM

The peak velocity achieved in simple harmonic motion, calculated as $\text{Max speed} = \omega A$, where $\omega$ is in $\text{s}^{-1}$ and $A$ is in $\text{m}$.

Maximum acceleration in SHM

The peak acceleration achieved in simple harmonic motion, calculated as $\text{Max acceleration} = \omega^2 A$, where $\omega$ is in $\text{s}^{-1}$ and $A$ is in $\text{m}$.

Time period of a mass-spring system

The time taken for one full oscillation in a simple harmonic mass-spring system, calculated as $T = 2\pi \sqrt{\frac{m}{k}}$, where $T$ is in $\text{s}$, $m$ is in $\text{kg}$, and $k$ is in $\text{N}\,\text{m}^{-1}$.

Time period of a simple harmonic pendulum

The time taken for one full oscillation of a simple pendulum, calculated as $T = 2\pi \sqrt{\frac{l}{g}}$, where $T$ is in $\text{s}$, $l$ is in $\text{m}$, and $g$ is in $\text{m}\,\text{s}^{-2}$.

Free vibrations

Vibrations where a system oscillates freely at a frequency known as its natural frequency.

Forced vibrations

Vibrations where a driving force causes the system to vibrate at a different frequency from its natural frequency.

Resonance

The state occurring when a driving force most efficiently transfers energy to a vibrating system, resulting in a phase difference of $\frac{\pi}{2}\text{ radians}$ between the driver and the oscillations.

Damping

The phenomenon that occurs when an opposing force dissipates energy from an oscillating system to the surroundings.

Critical damping

A form of damping that reduces the amplitude of an oscillation to zero in the quickest possible time.

Overdamping

A condition occurring when the damping force is too strong, causing the system to return to equilibrium slowly without oscillating.

Underdamping

A condition occurring when the damping force is too weak, causing the system to oscillate with exponentially decreasing amplitude.

Greater damping effects

For an oscillation subject to greater damping, the amplitude is lower across all frequencies due to greater energy losses, and the resonant peak becomes broader.