Understanding Rolling Without Slipping

The question asks about the velocity of the specific point on a circular object that is in contact with the horizontal floor. The object is rolling without slipping, and its center is moving with a velocity V.

Analyzing the Motion Components

Rolling motion without slipping can be broken down into two simultaneous components:

• Translational Motion: The center of the circular object moves horizontally with a velocity V. Every point on the object shares this translational velocity.
• Rotational Motion: The object rotates around its center with an angular velocity, let's denote it as ω.

Calculating Velocity at the Point of Contact

To find the net velocity of the point of contact (the point momentarily touching the floor), we need to combine the effects of both translational and rotational motion:

1. Translational Velocity Component: The point of contact, being part of the object, inherits the velocity of the center. So, it has a translational velocity of V in the direction of the object's overall motion.
2. Rotational Velocity Component: Due to the rotation, the point of contact has a tangential velocity. The magnitude of this tangential velocity is given by the formula v_t = ωr, where r is the radius of the object. This tangential velocity is directed opposite to the direction of motion (i.e., backward) relative to the center.
3. Condition for Rolling Without Slipping: The key condition for rolling without slipping is that the linear velocity of the center (V) and the angular velocity (ω) are related by the equation V = ωr.
4. Combining Components: Using the relationship V = ωr, we can see that the magnitude of the tangential velocity at the point of contact is equal to the velocity of the center, i.e., v_t = V. This velocity is directed backward.

Determining the Net Velocity

The total velocity of the point of contact is the vector sum of the translational velocity and the rotational velocity:

Velocity_{contact point} = Velocity_translation + Velocity_rotation

Substituting the values and directions:

Velocity_{contact point} = V (forward) + V (backward)

Velocity_{contact point} = V - V

Velocity_{contact point} = 0

Conclusion

Therefore, the point of contact between the circular object and the floor is instantaneously at rest. This is the defining characteristic of rolling without slipping.