Engr Help

Kinematics of a Particle in a Plane (using polar description)

Example: Sally and John are using a rope (as shown below) to pull a box along the ground. They

are able to move to the right at a speed of v_rope. Find the speed of the box in terms of v_rope and θ.

Strategy: Apply the definition of velocity in polar coordinates and use a coordinate transformation

between x˗y and r˗θ.

Solution: v_box = v_box i = dr/dt e_r + r dθ/dt e_θ Here dr/dt = v_rope

Now relate i to e_r and e_θ (coordinate transformation)

i = e_r cos θ* ˗ e_θ sin θ* where θ* = π/2 ˗ θ ; i = e_r sin θ ˗ e_θ cos θ

v_box (e_r sin θ ˗ e_θ cos θ ) = dr/dt e_r + r dθ/dt e_θ

So v_box sin θ = dr/dt = v_rope and finally v_box = v_rope csc θ (result)