A uniform disc is spun with an angular velocity vecω and simultaneously projected with a linear velocity v towards left on a plank, while the plank moves towards right with a constant velocity 2 v. If the disc rolls without sliding on the plank just after its spinning, the magnitude of vecω (in radian/s) is. (Take v=3 m / s , R=1 m ).

Question

A uniform disc is spun with an angular velocity vecω and simultaneously projected with a linear velocity v towards left on a plank, while the plank moves towards right with a constant velocity 2 v. If the disc rolls without sliding on the plank just after its spinning, the magnitude of vecω (in radian/s) is. (Take v=3 m / s , R=1 m ). (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

For pure rolling of the disc on the plank,

v_{P} = v_{Q}

where

v_{Q} = 2 v \hat{i}

This gives

v_{P} = 2 v

Since,

v_{P} = v_{PC} + v_{C}

Substituting

v_{P} = 2 v \hat{i}

and

v_{C} = - v \hat{i}

We have

v_{PC} = 3 v \hat{i}

Then using the formula

v_{PC} = ω_{PC} \times r_{PC}

Where

r_{PC} = - R \hat{j}

We can find

ω_{PC} (= ω)

= \frac{3 v}{R} \hat{k} = \frac{3 \times 3}{1} \hat{k} = 9 \hat{k} r a d / sec