The figure shows a system consisting of (i) a ring the outer radius 3R rolling clockwise without slipping on a horizontal surface with angular speed `omega` and (ii) an inner disc of radius 2R rotating anti clockwise with angular speed `omega//2.` The ring and disc are separted. The point P on the inner disc is at a distance R from the origin, where OP makes an angle of `30^@` with the horizontal. Then with respect to the horizontal surface,

Options

A. The point (O) has a linear velocity ` 3 R omega hat i`
B. The point (P) has a linear velocity ` (11)/4 R pmega hat I + (sqrt 3)/4 R omega hat k`
C. The point (P) has a linear velocity ` (13) /4 R omega hat I - (sqrt 3)/4 R omega hat k`
D. The point (P) has a linear velocity ` (3- (sqrt 3)/4 ) R omega hat I + 1/4 R omega hat k`

Answer

A, B

Explanation

As is clear from Fig. 2 (CF) . 72,
.
Iinear velocity ofn point ` O = r omega`
` vec v_0 = ( 3 R) omega hat i`,
along positive directin of x-axis.
Choice (a) is correct.
Lineaer velocity of point ` Pw.r.t. point (0) is `
` vec _(P_0)= vec _P- vec v_0` or ` vec v_p = vec _0 + vec v_(P_0)`
` = (3 R omega 0 hat i + (omega)/4 (-hat i) +( omega R)/2 (sqrt 3)/2 hat k`
`= (3 R omega ) hat i + ( omega) hat i + (omega hat i + ( sqrt R)/2 (s qrt 3 )/2 hat k`
` vec v_p = (11)/4 R omega hat i + ( sqrt 3)/4 R omega hat k`.
Choslces (a) and (b) are correct.