Motion in two dimensions, in a plane can be studied by expressing position, velocity and acceleration as vectors in cartesian coordinates `A=A_(x)hat(i) + A_(y)hat(j)`, where `hat(i)` and `hat(j)` are unit vector along x and y-directions, respectively and `A_(x)` and `A_(y)` are corresponding components of A. Motion can also be studied by expressing vectors in circular polar coordinates as `A= A_(r)hat(r) + A_(theta) hat(theta)`, where `hat(r)=(r)/(r)=cos theta hat(i)+sin theta hat (j)` and `hat(theta)=-sin theta hat(i) + cos theta hat (j)` are unit vectors along direction in which r and `theta` are increasing.
(a) Express `hat(i)` and `hat (j) ` in terms of `hat(r) ` and `hat (theta)`.
(b) Show that both `hat(r)` and `hat(theta)` are unit vectors and are perpendicular to each other.
(c) Show that `(d)/(dt)(hat(r))= omega hat(theta)`, where `omega=(d theta)/(dt)` and `(d)/(dt) (hat(theta))=-thetahat(r)`.
(d) For a particle moving along a spiral given by `r=a theta hat(r)`, where a = 1 (unit), find dimensions of a .
(e) Find velocity and acceleration in polar vector representation for particle moving along spiral described in (d) above.

Explanation

(a) Given, unit vector `" " hat(r)=cos theta hat(i) + sin theta hat(j) " " ...(i)`
`" " hat (theta)=-sin theta hat(i) + cos theta hat (j) " " ...(ii)`
Multiplying Eq. (i) by `sin theta` and Eq. (ii) with `cos theta` and adding
`hat(r)sin theta + hat (theta)cos theta=sin theta . cos theta hat(i) + sin^(2)theta(j) + cos^(2)theta hat(j) - sin theta. cos theta hat(i)`
`=hat(j) (cos^(2)theta + sin^(2)theta)=hat(i)`
`rArr " " hat(r)sin theta + hat(theta)cos theta=hat(j)`
By Eq. `(i) xx cos theta-"Eq."(ii) xx sin theta`
`n(hat(r)cos thet a-hat(theta)sin theta)=hat(i)`
(b) `hat(r).hat(theta)=(cos theta hat(i) + sin theta hat(j)).(-sin thetahat(i)+cos thetahat(j))=-cos theta.sin theta+sin theta.cos theta =0`
`rArr " " theta=90^(@)` Angle between `hat(r)` and `hat(theta)`.
(c) Given, `hat(r)=cos thetahat(i)+sin theta hat(j)`
`(dhat(r))/(dt)=(d)/(dt)(cos theta hat (i)+sin theta hat (j))= - sin theta. (d theta )/(dt) hat (i) + cos theta. (d theta)/(dt) hat (j)`
`= omega [-sin theta hat (i) + cos theta hat (j)] " " [ :' theta = (d theta)/(dt)]`
(d) Given , `r=a theta hat (r)`, here, writing dimensions `[r]=[a][theta][hat(r)]`
`rArr " " L=[a] % 1 rArr [a] =L =[M^(0)L^(1)T^(0)]`
(e) Given, a = 1 unit `r= theta hat (r) = theta [ cos theta hat (i) + sin theta hat (j) ]`
Velocity, ` " " v=(dr)/(dt)=(d theta)/(dt) hat (r) + theta (d)/(dt) hat (r) = (d theta)/(dt) hat (r) + theta (d)/(dt) [ cos theta hat (i) + sin theta hat (j) ]`
`=(d theta)/(dt) hat (r) + theta [(-sin theta hat (i) + cos theta hat (j)) (d theta)/(dt)]`
`=( d theta)/(dt) hat (r) + theta hat (theta) omega = omega hat (r) + omega theta hat ( theta)`
Acceleration, `" " a=(d)/(dt) [ omega hat (r)+omegathetahat(theta)]=(d)/(dt) [(d theta)/(dt) hat(r)+(d theta)/(dt) ( thetahat(theta))]`
`=(d^(2)theta)/(dt^(2))hat(r)+(d theta)/(dt).(d hat (r))/(dt)+(d^(2)theta)/(dt^(2))theta hat (theta)+(d theta)/(dt)(d)/(dt)(theta hat (theta))`
`=(d^(2)theta)/(dt^(2))hat(r) + omega [-sin theta hat (i) + sin theta hat (j)]+ (d^(2)theta)/(dt^(2))theta hat ( theta)+ (omega d)/(dt)( theta hat ( theta))`
`=(d^(2)theta)/(dt^(2))hat (r)+ omega^(2) hat ( theta)+ (d^(2)theta)/(dt^(2))xxtheta hat( theta)+ omega^(2)hat ( theta)+ omega^(2) theta (-hat(r))`
`((d^(2)theta)/(d t v^(2))- omega^(2))hat(r) + (2 omega^(2)+ (d^(2)theta)/(dt^(2))theta) theta`