In case of pure rolling what will be the velocity of point A of the ring of radius R

When we talk about pure rolling motion, we refer to a situation where an object rolls without slipping. In the case of a ring of radius R, point A, which is typically the point in contact with the ground, has a specific velocity that can be understood through the principles of rotational motion and linear motion.

Understanding Pure Rolling Motion

In pure rolling, the relationship between the linear velocity of the center of mass (V) and the angular velocity (ω) of the rolling object is given by the equation:

• V = R * ω

Here, V is the linear velocity of the center of the ring, R is the radius, and ω is the angular velocity. This equation indicates that the point of contact with the ground (point A) has a unique velocity due to the combination of both translational and rotational motion.

Velocity of Point A

At any instant, the velocity of point A can be analyzed as follows:

• When the ring rolls, point A is momentarily at rest relative to the ground. This is because, at the point of contact, the linear velocity of the ring (V) and the velocity due to rotation (which is directed opposite to the motion of the center) cancel each other out.
• The velocity of point A can be expressed as:
• Velocity of point A = V - R * ω

Since V = R * ω, substituting this into the equation gives:

• Velocity of point A = R * ω - R * ω = 0

This means that at the moment of contact with the ground, point A has a velocity of zero. It is not moving relative to the ground, which is a key characteristic of pure rolling motion.

Visualizing the Concept

To visualize this, imagine a bicycle wheel rolling along a flat surface. The bottommost point of the wheel (analogous to point A) is in contact with the ground and is not sliding; it is effectively "stationary" at that instant. As the wheel rolls forward, this point lifts off the ground and becomes the next point in contact, continuing the cycle.

Summary of Key Points

• In pure rolling motion, the velocity of the point of contact (point A) is zero.
• This occurs because the linear velocity of the center of the ring and the rotational velocity at point A cancel each other out.
• Understanding this concept is crucial for analyzing various rolling objects in physics.

In conclusion, the velocity of point A of the ring during pure rolling is zero at the instant it is in contact with the ground. This principle is fundamental in mechanics and helps explain the behavior of rolling objects in various contexts.