A uniform disc is spun with an angular velocity `omega` and simultaneously projected with a linear velocity `v` towards left on a plank, while the plank moves towards right with a constant velocity `2v`. If the disc rolls without sliding on the plank just after its spinning, find the magnitude of `vecomega`. (Take `v = 3 ms^(-1),R=1m`)

Answer

`9rad//s`

Explanation

For pure rolling of the disc on the plank

`vecv_(P)=vecv_(Q)`, where `vecv_(Q)=2vhati`
This gives `vecv_(P)=2vecv`
Since `vecv_(p)=vecv_(PC)=vecv_(C)`
Substituting `vecv_(P)=2vhati` and `vecv_(C)=-vecvhati`
we have`vecv_(PC)=3vhati`
Then using the formula `vecv_(PC)=vecomega_(PC)xxvecr_(PC)`
Where `vecr_(PC)=-Rhatj`
we can find `vecomega_(PC)(=vecomega)=(3v)/r hatk`