Definitions

1. Angular Speed ($ω$)

• Definition: The rate of change of angular displacement, representing how fast an object rotates.
• Formula: $\omega = 2\pi f$ or $\omega = \frac{\theta}{t}$ , where $\theta$ is the angle in radians, $t$ is time, and $f$ is the frequency.

2. Centripetal Acceleration

• Definition: The acceleration of an object moving in a circular path, directed towards the centre of the circle.
• Formula: $a = \frac{v^2}{r} = \omega^2 r$, where $v$ is linear speed, $r$ is radius, and $\omega$ is angular speed.
• Explanation: Even though the object's speed is constant, its direction changes continuously, requiring an inward acceleration.

3. Centripetal Force

• Definition: The force that keeps an object moving in a circular path, directed towards the centre of the rotation.
• Formula: $F = \frac{mv^2}{r} = m\omega^2 r$

4. Critical Damping

• Definition: The level of damping that brings a system back to equilibrium in the shortest possible time without further oscillation.
• Explanation: Critical damping is used in systems where rapid return to rest is needed, such as car shock absorbers.

5. Damping

• Definition: The process of energy dissipation in an oscillating system, leading to a gradual decrease in amplitude.
• Explanation: Damping occurs due to forces like friction or air resistance. It can slow or stop oscillations, depending on the damping level.

6. Forced Vibrations

• Definition: Oscillations that occur when an external force drives the system at a certain frequency.
• Explanation: The amplitude depends on the driving frequency; at high frequencies, amplitude is low, while at low frequencies, it increases.

7. Free Vibrations

• Definition: Natural oscillations that occur without an external force driving the motion.
• Explanation: A pendulum, if displaced and released, will oscillate at its natural frequency without any continuous external force.

8. Overdamping

• Definition: A damping level where the system is damped beyond the critical point, preventing oscillations but slowing the return to equilibrium.
• Explanation: In overdamped systems, the object takes longer to reach equilibrium compared to critically damped systems.

9. Radian

• Definition: A unit of angular measurement.
• Explanation: One radian is the angle formed when the arc length is equal to the radius of the circle. There are $2\pi$ radians in a full circle.

10. Resonance

• Definition: Occurs when the frequency of a driving force matches the natural frequency of the system, leading to a large increase in amplitude.
• Explanation: Resonance is beneficial in some applications, such as tuning musical instruments, but can be destructive in structures if not controlled (e.g., bridge oscillations).

11. Simple Harmonic Motion (SHM)

• Definition: Motion in which the acceleration of an object is directly proportional to its displacement and directed towards a fixed point.
• Formula: $a = -\omega^2 x$, where $x$ is displacement and $\omega$ is angular frequency.

12. Underdamping

• Definition: A type of damping where the system continues to oscillate while the amplitude gradually decreases over time.