Tru Physics

Simple Harmonic Motion

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Tru Physics
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Introduction

Simple Harmonic Motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. It is an important concept in physics due to its wide applicability in various phenomena such as pendulums, spring-mass systems, and oscillations in electrical circuits.

Equation of Motion

The differential equation governing SHM is given by:

\dfrac{d^2x}{dt^2} = -\omega^2x

where x is the displacement, t is the time, and \omega is the angular frequency of the motion. The solution to this equation is:

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

where A is the amplitude, \omega is the angular frequency, and \phi is the phase angle.

Velocity and Acceleration

The velocity and acceleration of an object undergoing SHM can be derived from the displacement as follows:

Velocity: v(t) = \dfrac{dx}{dt} = -A\omega\sin(\omega t + \phi)

Acceleration: a(t) = \dfrac{d^2x}{dt^2} = -A\omega^2\cos(\omega t + \phi)

Energy in SHM

The total mechanical energy of a system undergoing SHM is the sum of its kinetic and potential energy, and it remains constant if there’s no damping. The energy is given by:

E = \dfrac{1}{2}m\omega^2A^2

where m is the mass of the object and A is the amplitude of the motion.

Damped and Forced Oscillations

Real-world oscillations often involve damping (energy loss) and/or external forcing. Damped oscillations gradually decrease in amplitude, while forced oscillations can reach a steady state of constant amplitude called resonance, at a specific forcing frequency.

Applications

Simple Harmonic Motion is found in many areas of physics and engineering. Mechanical vibrations, electrical oscillations, wave motion, and quantum mechanics all involve aspects of SHM. Understanding this fundamental type of motion is crucial for the design of various systems, from vehicle suspensions to electrical circuits to buildings that can withstand earthquakes.

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