A solid ball, initially at rest, is given a sharp impulsc by...

A solid ball, initially at rest, is given a sharp impulsc by a cuc. The cuc is held horizontally a distanec $h$ above the central linc as in figure. The cue imparts a specd $v_{0}$ to the ball. It rolls and slides while moving forward and acquires a final speed of $(9 \pi) v_{\circ}$. Show that $h=(4 / 5) R$ where $R$ is the radius o\lceil the ball.

Key Concepts

Rolling Motion and the No?Slip Condition

Rolling motion combines both translational and rotational movements, and the no?slip condition (v = ?R) is the relationship that must be fulfilled for pure rolling. This condition ensures that the point of contact with the surface does not skid, meaning the linear velocity of the center of mass is directly related to the angular velocity. Analyzing problems involving rolling typically involves applying this constraint to relate different aspects of the motion.

Friction in Rolling and Sliding Transitions

Friction is essential in transitioning a body from simply sliding to rolling without slipping. When an object is initially given an impulse and both translation and rotation occur unevenly, friction acts to adjust the rotational and translational speeds until the no?slip condition is met. Thus, friction not only opposes unwanted sliding but also helps to develop the rotational motion necessary for pure rolling.

Impulse and Angular Momentum

When a force is applied over a very short time interval (an impulse), it can change both the linear momentum and the angular momentum of a rigid body. In problems where the impulse is applied away from the center of mass, the resulting ‘off?center’ impulse produces a torque, thereby imparting both translational and rotational motions. This concept is crucial for analyzing how a sudden push can simultaneously set an object in linear motion while also causing it to spin.

Moment of Inertia

The moment of inertia quantifies an object's resistance to changes in its rotational motion and depends on the mass distribution relative to the axis of rotation. For a solid sphere, the specific formula (usually I = (2/5)mR²) plays a vital role in linking the applied torque from an impulse (or force) to the resulting angular acceleration. Understanding the moment of inertia is key to analyzing the rotational dynamics of rolling objects.

Please give Ace some feedback