ENGR 215 ~ Dynamics Section 12.8

Polar Coordinates

r

r

rd

0dt

r uu

Velocity in Polar Coordinates

r r

r

v vdr

v rdt

dv r r

dt

v u u

Motion in a straight line

Lecture Example 1: Uniform Circular Motion

r = C1

θ = C2 tDetermine the velocity of the particle.

Acceleration in Polar Coordinate

2r

2r

(r r ) (r 2 r )

a (r r )

a (r 2 r )

a u u

Determine the acceleration of the particle.

Lecture Example 2: Uniform Circular Motion

r = C1

θ = C2 t

Lecture Example 3: Determine the speed of the slider blocks at θ = 120º.

Lecture Example 4: The car is traveling at 30m/s. Determine the angular rate at which the camera must turn to follow the car at the instant where θ=30º.

Lecture Example 5: The boy slides down the slide at a constant speed of 2 m/s. The slide forms a helix, defined by r = 1.5 m and z = -θ / π. Determine the angular velocity about the z-axis and the magnitude of the acceleration.

Lecture Example 6: At time, t = π seconds determine the velocity of the particle, and determine the tangential and normal components of the acceleration. Draw and label the acceleration vectors on the graph below. The position of the particle is expressed in meters. Trigonometric functions should be evaluated in radians.

-4

-3

-2

-1

0

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 0 1 2 3 4

r(t) = t 0 < t < 2 π sec

θ(t) = π/2 – t

Spiral of Archimedes