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Related Concept Videos

Relation Between Moment of a Force and Angular Momentum01:21

Relation Between Moment of a Force and Angular Momentum

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In the realm of spinning tops, the application of force at a distance from the center produces torque, a pivotal factor that alters the angular momentum of the top, thereby inducing its rotation. The concept of moment, akin to linear force in rotation, quantifies how a force acting upon an object initiates rotational motion. Angular momentum serves as the rotational counterpart to linear momentum, representing an object's inherent tendency to persist in its rotational state.
The temporal...
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Euler Equations of Motion01:19

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Imagine a rigid body that is rotating at an angular velocity of ω within an inertial frame of reference. Along with this, picture a second rotating frame that is attached to the body itself. This frame moves along with the body and possesses an angular velocity of Ω. The total moment about the center of mass is calculated by adding the rate of change of angular momentum about the center of mass in relation to the rotating frame and the cross-product of the body's angular velocity...
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Angular Momentum about an Arbitrary Axis01:11

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Imagine a rigid body with a mass denoted as 'm', which has its center of mass at point G and is rotating around an inertial reference frame. The angular momentum at an arbitrary point P can be calculated by taking the cross product of the position vector and linear momentum vector for each individual mass element.
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Principle of Angular Impulse and Momentum01:23

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The angular impulse and momentum principle provides insights into how forces applied at a distance from an object's rotational axis influence its angular velocity. It builds upon the crucial relationship between the moment of force and angular momentum. By integrating this equation, substituting the limits for the initial and final times, a comprehensive expression representing the angular impulse and momentum principle is derived.
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Angular Momentum and Principle Axes of Inertia01:09

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The concept of angular momentum for a solid structure is illustrated as the cumulative result of the cross-product of the position vector of the mass element and the cross-product of the body's angular velocity with the position vector.
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Angular Momentum: Single Particle01:10

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Angular momentum is directed perpendicular to the plane of the rotation, and its magnitude depends on the choice of the origin. The perpendicular vector joining the linear momentum vector of an object to the origin is called the “lever arm.” If the lever arm and linear momentum are collinear, then the magnitude of the angular momentum is zero. Therefore, in this case, the object rotates about the origin such that it lies on the rim of the circumference defined by the lever arm...
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Isotopic Effect in Double Proton Transfer Process of Porphycene Investigated by Enhanced QM/MM Method
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Generalization of adding angular momenta and circular potential in quaternionic quantum mechanics.

R Deepika1, K Muthunagai1

  • 1School of Advanced Sciences, Vellore Institute Of Technology, Vandalur-Kelambakkam Road, Chennai, 600 127, Tamil Nadu, India.

Heliyon
|February 19, 2024
PubMed
Summary

This study explores quaternions, an extension of complex numbers, for advanced mathematical and physics applications. Quaternions offer novel solutions in areas like angular momentum and signal processing, yielding results in Bessel functions.

Keywords:
Angular momentumAnti-Hermitian operatorCircular potentialQuaternions

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Area of Science:

  • * Mathematics
  • * Physics
  • * Signal Processing

Background:

  • * Complex numbers, using the imaginary unit 'i', solve equations with negative square roots.
  • * Quaternions extend complex numbers with 'j' and 'k' imaginary units, forming non-commutative four-component numbers.
  • * Complex numbers commonly represent wave functions in quantum mechanics, while quaternions have shown promise in areas like spin and Dirac equations.

Purpose of the Study:

  • * To investigate the application of quaternions in generalizing the addition of angular momenta.
  • * To explore the digital signal processing of quaternionic functions.
  • * To analyze the circular potential of a particle in real Hilbert space using quaternions.

Main Methods:

  • * Employed quaternionic algebra to address the generalization of adding angular momenta.
  • * Applied quaternionic functions to digital signal processing techniques.
  • * Utilized quaternions to model a particle's circular potential in a real Hilbert space.

Main Results:

  • * Derived generalized solutions for adding angular momenta using quaternions.
  • * Developed digital signal processing methods for quaternionic functions.
  • * Obtained solutions for the circular potential problem in terms of Bessel functions, using a quaternionic approach.

Conclusions:

  • * Quaternions provide a powerful framework for extending mathematical and physical concepts.
  • * The study successfully demonstrated quaternionic solutions in angular momentum, signal processing, and quantum mechanics.
  • * Bessel functions emerged as key components in the derived quaternionic solutions for specific physical problems.