Name: radial and tangential motion in polar form
Uploaded: 2024-03-07T13:22:15.000Z
Duration: 1 min 27 s
Channel: JoVE

The curvilinear motion of a particle can be described using the polar coordinates system.
The radial coordinate, denoted as &#39;r,&#39; extends outward from the fixed origin to the particle. The angular coordinate, &#39;&#952; (theta),&#39; measured in radians, is the counterclockwise angle between a fixed reference line and the radial line connecting the origin to the point.
The particle&#39;s position can be expressed using a unit vector along the radial direction. Differentiating the position of the object with time gives the velocity.
Here, the first term is linear velocity along the radial direction, and the second term is the transverse velocity component of the object. These two components of velocity are always perpendicular to each other.
The time derivative of the velocity expression gives the acceleration. The rate of change of the angular unit vector equals the negative product of angular velocity with the radial unit vector.
Here, the second derivative of the angular coordinate is the angular acceleration of the object. Substituting the terms gives the expression for acceleration having the components perpendicular to each other.&#160;

curvilinear motion: polar coordinates

Curvilinear Motion: Polar Coordinates

In polar coordinates, the motion of a particle follows a curvilinear path. The radial coordinate symbolized as &#39;r,&#39; extends outward from a fixed origin to the particle, while the angular coordinate, &#39;&#952;,&#39; measured in radians, represents the counterclockwise angle between a fixed reference line and the radial line connecting the origin to the particle.
The particle&#39;s location is described using a unit vector along the radial direction. Deriving the particle&#39;s position with respect to time provides its velocity. This velocity is comprised of two components: the first is the linear velocity along the radial direction, and the second is the tangential velocity perpendicular to the radial direction.
The time derivative of the velocity yields the acceleration. The angular unit vector&#39;s rate of change is the negative product of the angular velocity and the radial unit vector. The second derivative of the angular coordinate represents the angular acceleration of the particle. Analogous to velocity, both components of acceleration are mutually perpendicular. In summary, the polar coordinate system elegantly captures the intricacies of curvilinear motion, unveiling the interplay between radial and tangential dynamics.

The curvilinear motion of a particle can be described using the polar coordinates system.
The radial coordinate, denoted as &#39;r,&#39; extends outward ...

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Mechanical Engineering

Unit 1

Kinematics of a Particle

KEY TERMS AND CONCEPTS

radial and tangential motion in polar form

Polar coordinates describe curvilinear motion using a radial distance and an angle. The radial coordinate, r, extends from a fixed origin to the particle. The angular coordinate, θ, is measured in radians. It gives the counterclockwise angle between a fixed reference line and the radial line to the particle.

The particle’s position is written with a unit vector in the radial direction. Taking the time derivative of that position gives the velocity. That velocity has two parts. One is radial, or linear motion along the radial direction. The other is tangential motion, which is perpendicular to the radial direction.

Taking the time derivative of velocity gives the acceleration. The rate of change of the angular unit vector equals the negative product of the angular velocity and the radial unit vector. The second derivative of the angular coordinate gives the particle’s angular acceleration. Like velocity, acceleration in polar form has two perpendicular components. This makes polar coordinates a clear way to describe the radial and tangential parts of curvilinear motion.

Radial and Tangential Motion in Polar Form

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curvilinear motion: normal and tangential components

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Polar coordinates describe curvilinear motion using a radial distance and an angle. The radial coordinate, r, extends from a fixed origin to the particle. The angular coordinate, θ, is measured in radians. It gives the counterclockwise angle between a fixed reference line and the radial line to the particle. The particle’s position is written with a unit vector in the radial direction. Taking the time derivative of that position gives the velocity. That velocity has two parts. One is radial,...

key terms and concepts

Learn how polar coordinates describe curvilinear motion with radial distance, angle, velocity, and acceleration broken into radial and tangential parts.

Radial and Tangential Motion in Polar 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.

Radial and Tangential Motion in Polar Form

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