A spinning spherical ball having angular velocity (vθ / 2 r) is projected on a horizontal rough

step 1 Hello everyone. So for this question the F, Ext is given that is zero. So linear momentum of the

A spinning spherical ball having angular velocity (vθ / 2 r)...

A spinning spherical ball having angular velocity $\left(v_{\theta} / 2 r\right)$ is projected on a horizontal rough surface with velocity $v_{i}$. Calculate the velocity of centre of mass when slipping ceases.

Key Concepts

Rolling without Slipping Condition

This condition occurs when the peripheral speed of a rotating object equals the linear speed of its center of mass, meaning that v = ?r. It signifies that there is no relative motion between the point of contact and the surface, which is a key requirement in problems involving rolling motion after slipping ceases.

Frictional Forces in Rotational Motion

Friction plays a dual role by affecting both the translational and rotational movements of an object. When an object is both spinning and translating on a rough surface, kinetic friction works to decelerate the slipping motion while simultaneously providing a torque that alters the object's angular velocity, ultimately driving the system toward the rolling without slipping condition.

Translational and Rotational Dynamics

The dynamics of a rolling body are governed by both linear and angular forms of Newton’s second law. The translational motion is described by F = ma, while the rotational motion is governed by ? = I?. The interplay between these two sets of equations is essential to model how an object transitions from slipping to pure rolling motion.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion. For a sphere, it is typically represented as a fraction of mr² (commonly 2/5 m r²), and it determines how the torque due to friction affects the angular acceleration. This property is critical in establishing the relationship between the angular deceleration and the subsequent achievement of the rolling without slipping condition.

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