Tru Physics

Oscillations

—

by

Tru Physics
in

Introduction

Oscillations are repetitive variations, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. In physics, oscillations include the movements of a pendulum and the vibrations of atoms.

Simple Harmonic Motion

Simple harmonic motion (SHM) is a type of oscillatory motion in which the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. It can be described by the equation:

m\dfrac{d^2x}{dt^2} = -kx

where m is the mass of the object, x is the displacement from the equilibrium position, and k is the spring constant. The solution to this differential equation is:

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

where A is the amplitude, \omega = \sqrt{k/m} is the angular frequency, t is time, and \phi is the phase constant.

Energy in Simple Harmonic Motion

In SHM, energy oscillates between potential energy and kinetic energy, but the total energy remains constant. The total energy E of an object in SHM is given by:

E = \dfrac{1}{2}kA^2

where A is the amplitude of the motion and k is the spring constant.

Damping

Damping is an influence within or upon an oscillatory system that has the effect of reducing the amplitude of the oscillations. A damping force can often be approximated as being proportional to the velocity of the object:

F = -b \dfrac{dx}{dt}

where b is the damping constant.

Forced Oscillations and Resonance

A forced oscillation occurs when an external periodic force drives a system to oscillate with a frequency equal to the frequency of the force. Resonance occurs when the frequency of the driving force matches the natural frequency of the system, leading to large oscillations.

Oscillations in Multiple Dimensions: Waves

When oscillations occur in a medium with an extended dimension, they can form waves. A sinusoidal wave is described by the wave equation:

\dfrac{\partial^2y}{\partial t^2} = v^2 \dfrac{\partial^2y}{\partial x^2}

where y(x, t) is the displacement of the medium, v is the speed of the wave, t is time, and x is position.

Nonlinear Oscillations and Chaos

In many real-world systems, the restoring force is not proportional to the displacement, leading to nonlinear oscillations. These can have complex behaviors, including chaos, where small changes in initial conditions can lead to large differences in outcomes.

Conclusion

Oscillations are a fundamental concept in physics with wide-ranging applications, from the design of clocks and musical instruments to the study of atomic vibrations and electromagnetic waves. Understanding the principles of oscillations is key to many areas of science and engineering.

Do you prefer video lectures over reading a webpage? Follow us on YouTube to stay updated with the latest video content!

Want to study more? Visit our Index here!

Comments

Have something to add? Leave a comment!