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The Polar Description - Kinematics

Key Concepts: Description of motion of a particle using polar description starts with the position

vector from the origin to the particle as in calculus. The first derivative of the position vector gives

the velocity of the particle and the second derivative gives its acceleration. Note that the direction

of both unit vectors , e_r , in the radial direction and e_θ , in the transverse direction change with time.

In a Nut Shell: Start with the position vector, r. Its derivative gives the velocity vector, v. Likewise

The derivative of the velocity vector, v, gives the acceleration vector, a.

r = r e_r = position vector of particle at point P at any time t; here e_r is the unit vector in

the radial direction and e_θ is the unit vector in the direction of increasing θ

v = dr /dt = (dr/dt) e_r + (r dθ/dt) e_θ = velocity vector of particle at point P at any time t

NOTE: dr/dt is the radial component of velocity and r dθ/dt is the transverse component

The magnitude of the velocity, |v|, is called speed. It is a scalar.

a = dv /dt = [d²r/dt² - r (dθ/dt)²] e_r + [ r d²θ/dt² + 2 (dr/dt)(dθ/dt)] e_θ

equals the acceleration vector of particle at point P at any time t

NOTE: [d²r/dt² - r (dθ/dt)²] is the radial component of acceleration and

[r d²θ/dt² + 2 (dr/dt)(dθ/dt)] is the transverse component of acceleration

The polar description of motion of a particle in a plane is useful when the path is some type of curve (curvilinear motion) in the plane although the rectangular description could also be used. Recall the transformation from rectangular to polar coordinates as follows

x = r cos θ , y = r sin θ NOTE: Both r and θ in general are functions of time.

Click here for examples. Click here to continue with intrinsic (normal-tangential) description.

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