A sphere is rolling without slipping on a fixed horizontal plane surface. In the figure, A is the point of contact, B is the centre of the sphere and C is its topmost point. Then :

Options

A. `vec(V)_(C^(-))vec(V)_(A=2)(vec(V)_(B^(-))vec(V)_(C ))`
B. `vec(V)_(C^(-))vec(V)_(B=)vec(V)_(B^(-))vec(V)_(A)`
C. `|vec(V)_(C^(-))vec(V)_(A)|=2|vec(V)_(B^(-))vec(V)_(C )|`
D. `|vec(V)_(C^(-))vec(V)_(A)|=4|vec(V)_(B)|`

Answer

C

Explanation

`vec(V)_(A)=V hat(i)+omega R(-hat(i)), vec(V)_(B)=V hat(i) , vec(V)_(C )=V hat(i)+ omega R hat(i)`
`vec(V)_(C )-vec(V)_(A)=2omega R hat(i)`
`2[vec(V)_(A)-vec(V)_(C )]=2[V(i)-V(i)-omega R(i)]=-2 omega R(i)`
Hence `vec(V)_(C )-vec(V)_(A)=-2(vec(V)_(B)-vec(V)_(C ))`