A sphere having radius 5 cm rolling purely on a horizontal surface with angular velocity 3 rads^-1, then what is the linear velocity for bottom-most point?

CONCEPT:

• Rolling motion: Rolling motion can be regarded as a combination of pure rotation and pure translation.
• In pure rolling motion, the point of contact (bottom-most point) moves with velocity V_p = velocity of point P = V
• V_p = (V’ - ωR) = V
• V’ = V_p – ωR

Where V’ = linear velocity of solid sphere

EXPLANATION:

• When an object is rolling on a plane without slipping, the point of contact of the object with the plane does not move.
• In pure rolling motion, there is no relative motion between body & floor on which it is rolling (No sleep).

V_{bottom point} = V_linear + (- V_rotational) = 0           

• Hence, net linear velocity for the bottom-most point in pure rolling is zero. Therefore option 4 is correct.

NOTE:

If there is no pure rolling then two types of sleep occur

1. Forward slip

2. Backward slip

1. Forward slip

Linear velocity ˃ linear velocity (rotation)

So,  V_net = V – (Rꙍ)_rotation

2. Backward slip

Linear velocity ˂ linear velocity (rotation)

So,  V_net = (Rꙍ)_rotation – V