Motion in two dimensions, in a plane can be studied by expressing position, velocity and acceleration as vectors in Cartesian co-ordinates A = A_{x} cap i + A_{y} cap j where cap i and cap j are unit vector along x and y directions, respectively and A_{x} and A_{y} are corresponding components of A (Fig. 4.9). Motion can also be studied by expressing vectors in circular polar co-ordinates as A = A_{r} cap r + A_{θ} cap θ where cap r = r/r cosθ sinθ and are unit vectors along direction in which ‘r’ and ‘θ ’ are increasing.

(a) Express cap i and cap j in terms of cap r and cap θ

(b) Show that both cap r and cap θ are unit vectors and are perpendicular to each other

(c) Show that (d/dt) (cap r) = *ω cap *θ where

(d) For a particle moving along a spiral given by r = aθ cap r , where a = 1 (unit), find dimensions of ‘a’.

(e) Find velocity and acceleration in polar vector represention for particle moving along spiral described in (d) above.