Tru Physics

Periodic Motion

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Introduction

Periodic motion, often associated with oscillatory or vibrational systems, is a fundamental concept in physics. This type of motion repeats itself in a regular cycle. Examples include the motion of a pendulum, the vibration of a guitar string, and the orbit of a planet around the Sun.

Basic Definitions

  • Periodic Motion: Motion that repeats itself over an equal interval of time, known as the period.
  • Period (T): The time taken for one complete cycle of the motion. It is usually measured in seconds (s).
  • Frequency (f): The number of cycles per unit of time. The unit for frequency is Hertz (Hz) and it is the reciprocal of the period: f = \frac{1}{T}.
  • Amplitude (A): The maximum displacement from the equilibrium position.

Simple Harmonic Motion (SHM)

The simplest type of periodic motion is Simple Harmonic Motion (SHM). It is the motion of a particle about a fixed point such that its acceleration is proportional to its displacement from the fixed point but in the opposite direction. A common example is the motion of a mass on a spring.

The displacement x(t) of a particle undergoing SHM is given by:

x(t) = A \cos(\omega t + \phi)

where

  • A is the amplitude,
  • \omega is the angular frequency (\omega = 2\pi f = \frac{2\pi}{T}),
  • t is time,
  • \phi is the phase constant, which determines where in its cycle the motion is at t = 0.

The velocity v(t) and acceleration a(t) of the particle are given by:

v(t) = -A \omega \sin(\omega t + \phi)

a(t) = -A \omega^2 \cos(\omega t + \phi)

Damped and Forced Oscillations

Real-world oscillators often involve damping (due to friction or other resistive forces) and external driving forces.

A damped oscillator’s motion is described by the differential equation:

m \dfrac{d^2x}{dt^2} + b \dfrac{dx}{dt} + kx = 0

where m is the mass, b is the damping constant, and k is the spring constant.

In forced oscillations, an external force is applied, leading to the differential equation:

m \dfrac{d^2x}{dt^2} + b \dfrac{dx}{dt} + kx = F_0 \cos(\omega_d t)

where F_0 is the amplitude of the external force, and \omega_d is the frequency of the external force.

Conclusion

Periodic motion is an essential concept in physics, with applications ranging from mechanics to optics to quantum mechanics. Its mathematical description leads to a rich variety of behavior, including resonance phenomena, wave propagation, and quantum oscillations.

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