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Motion in Two Dimensions

Quite often, we find a particle constrained to move in a plane, rather than in space in general. If we call this plane as XY plane, then position vector of the particle is given by,

r = x i + y j

because z = 0 always for the particle during motion. The spherical polar co-ordinates (r, θ, Ø) then reduce to circular polar co-ordinates (r, Ø); θ = 90˚ for the particle. That is, we define the position of the particle in terms of radial distance r from origin and angle Ø such that,

tan Ø = y/x                                       r = r sin Ø                (ii)

The displacement vector dr in circular polar co-ordinate is obtained by putting θ = 90˚ and d θ = 0

d r = d_r e_r + rd Ø e_Ø                                     (iii)

where unit vectors e_r, e_Ø are given by,

e_r = cos Ø i + sin Ø j                                  (iv)

e_Ø = –sin Ø i + cos Ø j                                                                 (v)