NoteTube

Motion of a particle in a plane in terms of planar polar coordinates
24:24

Motion of a particle in a plane in terms of planar polar coordinates

Engineering Mechanics

4 chapters6 takeaways11 key terms5 questions

Overview

This video introduces dynamics by explaining how to describe particle motion using planar polar coordinates, an alternative to Cartesian coordinates. It details the setup of the polar coordinate system, including the position vector and unit vectors (r-hat and phi-hat). The video then derives the expressions for velocity and acceleration in polar coordinates, highlighting how the changing nature of the unit vectors necessitates careful differentiation. Finally, it illustrates the application of these concepts with the example of circular motion, showing how polar coordinates simplify certain problems in dynamics.

How was this?

Save this permanently with flashcards, quizzes, and AI chat

Chapters

  • Dynamics describes the motion of particles, which requires a coordinate system to define position and velocity.
  • Newton's Second Law (F=ma) relates force, mass, and acceleration, which is the time derivative of velocity.
  • Cartesian coordinates (x, y) are familiar but can be cumbersome for certain types of motion, like circular paths.
  • Planar polar coordinates (r, phi) offer an alternative, particularly useful for rotational or circular motion.
Understanding different coordinate systems is crucial because the choice of system can significantly simplify the mathematical description and analysis of motion, especially for non-linear paths.
Describing a particle moving in a circle of radius R with angular velocity omega using Cartesian coordinates requires x(t) = R cos(omega*t) and y(t) = R sin(omega*t), where both x and y change with time. In polar coordinates, this motion is described by a constant radius R and an angle phi(t) = omega*t, simplifying the description as only the angle changes with time.
  • The position of a particle in planar polar coordinates is defined by its radial distance (r) from the origin and the angle (phi) its position vector makes with the x-axis.
  • The Cartesian coordinates (x, y) can be related to polar coordinates by x = r cos(phi) and y = r sin(phi).
  • The unit vectors in this system are r-hat (radial direction) and phi-hat (tangential direction, perpendicular to r-hat).
  • The unit vectors r-hat and phi-hat are expressed in terms of Cartesian unit vectors i and j: r-hat = cos(phi)i + sin(phi)j and phi-hat = -sin(phi)i + cos(phi)j.
  • Crucially, r-hat and phi-hat are not constant; they change direction as the particle's position (phi) changes.
Understanding the definitions and relationships of polar coordinates and their unit vectors is fundamental for deriving and interpreting the equations of motion in this system.
The unit vector r-hat, initially pointing along the positive x-axis when phi=0, rotates counter-clockwise as phi increases. At phi=90 degrees, r-hat points along the positive y-axis. This change in direction is a key characteristic that distinguishes polar coordinates from fixed Cartesian unit vectors.
  • The position vector in polar coordinates is r = r * r-hat.
  • Velocity (v = dr/dt) requires differentiating the position vector, accounting for the time-varying nature of both 'r' and 'r-hat'.
  • The derivative of r-hat with respect to time is dr-hat/dt = phi-dot * phi-hat, where phi-dot is the rate of change of the angle.
  • The velocity vector is expressed as v = r-dot * r-hat + r * phi-dot * phi-hat.
  • Geometrically, velocity has a radial component (r-dot) and a tangential component (r * phi-dot).
Deriving the velocity correctly in polar coordinates requires accounting for the change in direction of the radial unit vector, leading to a more complex but accurate representation of motion than simply r-dot.
For a particle moving in a circle of constant radius R, r is constant, so r-dot = 0. The velocity simplifies to v = R * omega * phi-hat, where omega is the angular velocity (phi-dot), representing the tangential speed R*omega in the tangential direction.
  • Acceleration (a = dv/dt) is found by differentiating the velocity vector v = r-dot * r-hat + r * phi-dot * phi-hat.
  • This differentiation involves terms like dr-hat/dt and d(phi-hat)/dt, which are related to phi-dot and r-dot.
  • The derivative of phi-hat with respect to time is d(phi-hat)/dt = -phi-dot * r-hat.
  • After differentiating and collecting terms, the acceleration vector is a = (r-double-dot - r * phi-dot^2) * r-hat + (r * phi-double-dot + 2 * r-dot * phi-dot) * phi-hat.
  • The terms r * phi-dot^2 and 2 * r-dot * phi-dot are often referred to as centripetal and Coriolis accelerations, respectively.
The acceleration expression in polar coordinates reveals components that are not present in Cartesian coordinates, such as the centripetal term (due to changing direction) and the Coriolis term (due to the interaction of radial and angular motion), which are essential for analyzing complex dynamics.
For a particle moving in a circle of constant radius R with constant angular velocity omega, r = R, r-dot = 0, r-double-dot = 0, and phi-dot = omega, phi-double-dot = 0. The acceleration simplifies to a = -R * omega^2 * r-hat, which is the familiar centripetal acceleration directed towards the center of the circle.

Key takeaways

  1. 1Planar polar coordinates (r, phi) are a powerful alternative to Cartesian coordinates for describing motion, especially circular or rotational motion.
  2. 2The unit vectors in polar coordinates (r-hat, phi-hat) are not fixed in direction and change as the particle's angular position changes.
  3. 3The velocity in polar coordinates has both a radial component (r-dot) and a tangential component (r * phi-dot).
  4. 4The acceleration in polar coordinates includes radial, tangential, centripetal (-r * phi-dot^2), and Coriolis (2 * r-dot * phi-dot) terms.
  5. 5The mathematical complexity of polar coordinates arises from the need to account for the changing directions of the unit vectors.
  6. 6Choosing the appropriate coordinate system can significantly simplify the analysis of physical phenomena.

Key terms

DynamicsCartesian CoordinatesPlanar Polar CoordinatesRadial Distance (r)Angle (phi)Unit Vector (r-hat)Unit Vector (phi-hat)Radial Velocity (r-dot)Angular Velocity (phi-dot)Centripetal AccelerationCoriolis Acceleration

Test your understanding

  1. 1Why are planar polar coordinates often preferred over Cartesian coordinates for describing circular motion?
  2. 2How does the time-dependence of the unit vectors r-hat and phi-hat affect the derivation of velocity and acceleration in polar coordinates?
  3. 3What are the physical interpretations of the terms in the acceleration equation for planar polar coordinates?
  4. 4How would you express the velocity of a particle moving radially outward from the origin in polar coordinates?
  5. 5Explain the difference between the radial and tangential components of velocity in polar coordinates.

Turn any lecture into study material

Paste a YouTube URL, PDF, or article. Get flashcards, quizzes, summaries, and AI chat — in seconds.

Summarize Your Own VideoSign up free

No credit card required

Motion of a particle in a plane in terms of planar polar coordinates | NoteTube | NoteTube