Sarthaks Test
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 Motion in two dimensions, in a plane can be studied by expressing position, velocity and acceleration as vectors in Cartesian co-ordinates A = Ax cap i + Ay cap j  where  cap i and cap j are unit vector along x and y directions, respectively and Ax and Ay are corresponding components of A (Fig. 4.9). Motion can also be studied by expressing vectors in circular polar co-ordinates as A = Ar cap r + Aθ 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.

 

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V = ω cap ωθ cap θ and a =

 

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