Name: particle motion in rectangular and cylindrical form
Uploaded: 2024-03-07T13:22:17.000Z
Duration: 1 min 21 s
Channel: JoVE

For a particle moving relative to an inertial frame, the equation of motion can be written using rectangular components. If motion is restricted to the x-y plane, only the first two equations apply.
Conversely, the equation of motion for a particle moving along a known curved path can be formulated in cylindrical components: radial, azimuthal, and axial, along respective unit vector directions.
The axial direction is perpendicular to the plane formed by the radial and azimuthal directions.
Here, the force along each component gives the acceleration along that particular component.
The acceleration of the particle along the radial component is the difference between the acceleration of the particle along the radial directions and the product of the radius and angular velocity squared.
The acceleration along the azimuthal component is the sum of the product of radius and angular acceleration and the product of the radial and angular velocity.
The acceleration along the axial direction corresponds to the change in speed of the particle along the vertical axis of the cylindrical system.

bewegingsvergelijking: rechthoekige coördinaten en cilindrische coördinaten

Equations of Motion: Rectangular Coordinates and Cylindrical Coordinates

Het begrijpen van de beweging van deeltjes is een fundamenteel aspect van de klassieke mechanica, en de keuze van het coördinatensysteem speelt een cruciale rol bij het ontrafelen van de complexiteit van hun dynamiek.

Wanneer een deeltje beweegt ten opzichte van een traagheidsframe, kunnen de bewegingsvergelijkingen worden uitgedrukt met behulp van rechthoekige componenten. Als de beweging beperkt is tot het x-y-vlak, kunnen de vergelijkingen met alleen de x- en y-coördinaten worden gebruikt om de wiskundige weergave te vereenvoudigen.

Wanneer deeltjes echter een gebogen pad volgen, wordt het cilindrische coördinatensysteem onmisbaar. Dit systeem introduceert radiale, azimutale en axiale componenten die zijn uitgelijnd met hun respectievelijke eenheidsvectorrichtingen en voegt een verticale dimensie toe aan de analyse, essentieel voor het vastleggen van de nuances van driedimensionale beweging. Binnen dit raamwerk bepaalt de kracht langs elke component de versnelling in de overeenkomstige richting. De radiale versnelling vertegenwoordigt bijvoorbeeld het verschil tussen de versnelling van het deeltje in de radiale richting en het product van zijn straal en hoeksnelheid. Omgekeerd is de azimutale versnelling een samenstelling van het product van straal en hoekversnelling gekoppeld aan het product van radiale en hoeksnelheid. Deze vergelijking verklaart de verandering in de positie van het deeltje, langs zijn gebogen traject en biedt waardevolle inzichten in de rotatieaspecten van zijn beweging. De axiale versnelling weerspiegelt de veranderingen in de snelheid van het deeltje langs de verticale as van het cilindrische systeem, waardoor inzicht wordt verkregen in de dynamiek van het deeltje in de ruimte.

Of het nu gaat om het benutten van de eenvoud van rechthoekige coördinaten of het omarmen van de extra dimensies van cilindrische coördinaten, elke benadering vergroot het begrip van hoe deeltjes bewegen en interageren met hun omgeving.

Bewegingsvergelijking: rechthoekige coördinaten en cilindrische coördinaten

For a particle moving relative to an inertial frame, the equation of motion can be written using rectangular components. If motion is restricted to the ...

equations-motion-rectangular-coordinates-cylindrical

Education

JoVE Core

Mechanical Engineering

Werktuigbouwkunde

Unit 1

Kinetics of a Particle: Force and Acceleration

Kinetica van een partikel: kracht en versnelling

kinetics-of-a-particle-force-and-acceleration

KEY TERMS AND CONCEPTS

BELANGRIJKE TERMEN EN CONCEPTEN

particle motion in rectangular and cylindrical form

Particle motion can be written in rectangular coordinates or cylindrical coordinates, depending on the path of the particle. In classical mechanics, the coordinate system you choose helps describe how a particle moves in an inertial frame.

When motion stays in the x-y plane, rectangular components are often enough. Using only x and y makes the equation easier to write and follow. This form works well for motion that does not need a third spatial direction.

Curved paths often call for cylindrical coordinates. This system uses radial, azimuthal, and axial components, and each one follows its own unit vector direction. The radial and azimuthal parts describe motion around the curved path, while the axial part adds the vertical dimension needed for three-dimensional motion.

In cylindrical coordinates, the force along each component determines the acceleration in that same direction. Radial acceleration depends on the particle’s radial motion and the product of its radius and angular velocity. Azimuthal acceleration combines radius, angular acceleration, radial motion, and angular velocity. Axial acceleration shows how the particle’s speed changes along the vertical axis. Together, these coordinate systems give a clearer picture of how particles move and interact with their surroundings.

Particle Motion in Rectangular and Cylindrical Form

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equations of motion: rectangular coordinates and cylindrical coordinates

Particle motion can be written in rectangular coordinates or cylindrical coordinates, depending on the path of the particle. In classical mechanics, the coordinate system you choose helps describe how a particle moves in an inertial frame. When motion stays in the x-y plane, rectangular components are often enough. Using only x and y makes the equation easier to write and follow. This form works well for motion that does not need a third spatial direction. Curved paths often call for...

equations of motion: normal and tangetial components

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Curved Motion: Tangential and Normal Forces

Curved motion is described by splitting a particle’s acceleration into tangential and normal parts. These two components help show how the particle moves along a curving path at a specific point. The tangential component lies along the curve. It shows how the speed changes with time. A positive tangential acceleration means the speed is increasing. A negative tangential acceleration means the speed is decreasing. The normal component points along the radial direction of the curve. It...

normal and tangetial components: problem solving

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Chair Reaction Forces in Curvilinear Motion

Chair reaction forces in curvilinear motion are found by using tangential and normal components of acceleration. A 70 kg man sits upright in a chair that is connected to a pin support by member BC. The member is 10 m long, and it makes a 45° angle with the horizontal at the instant of interest. The man has a speed of 5 m/s, and that speed is increasing at 1 m/s². The tangential acceleration is therefore 1 m/s². The normal acceleration is then determined from the tangential speed and the radius...

equation of motion: center of mass

Center of Mass and System Motion

The center of mass helps describe the motion of a system of particles. For a group with n particles, the equation of motion can be extended from a single particle to the whole system. Each particle has a net force on it that comes from both internal forces and external forces. When the motion of all particles is combined, the system follows one equation of motion. Forces between pairs of particles are internal forces. These forces act along the same line, have equal magnitude, and point in...

central-force motion

Central Force Motion and Areal Velocity

Central force motion describes the path of an object acted on by a force that points toward a fixed point, usually the origin. The force size depends on how far the object is from that point. For a mass m, the motion is written in polar coordinates, which use distance and angle instead of x and y position. In this system, the azimuthal component of force is zero. That means there is no force in the angular direction. When the equation of motion is rewritten and integrated, the product of the...

key terms and concepts

Learn how particle motion is written in rectangular and cylindrical coordinates, including radial, azimuthal, and axial acceleration components.

Particle Motion in Rectangular and Cylindrical Form | JoVE K12

JoVE Core Mechanical Engineering offers concise, easy-to-understand animated videos covering key topics such as equilibrium, forces, truss analysis, and structural mechanics. Explore their real-world applications, and witness these concepts in action through scientist-led experiments.

Particle Motion in Rectangular and Cylindrical Form

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