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 _A = 2 (vec v_B - vec v_C)`
B. ` vec v_c - vec _B = vec v_B - vec _A`
C. ` | vec _C - vec _A | = 2 | vec _B - vec _B- vec _C|`
D. `| vec _c - vec v_A | = 4 | vec _B|`

Answer

C

Explanation

Let ` ometga` be the angular velocity of rotation fo sphere.
` vec v_A, vec _B, vec_C` be the veloities at ` A, B` and ` C` respectively. Then (A) is at rest, i.e. ` | vec v_A| =0`.
.
` | vec v_C| = 2 omega R` about ` A, | vec _B| = omega R` about (A) `
:. | vec _C -vec _C| = 2 omega R- omega R= omega R`
` |vec _B- vec_A| = omega R- 0 = omega R`
` :. | vec _C-vec v_B|=| vec_B-vecv_A|`
Also , ` | vec_C-vec v_S| = (2 omega R-0)= 2 omega R`
` | vec _C -vec v_a| = | omega R- 2 omga R| =| omega R|`
:. ` | vec v_c - vec v_A| = 2 |vec _B vec v_C|` .