Suppose a solid disk of radius R is given an angular speed...

Suppose a solid disk of radius $R$ is given an angular speed $\omega_{i}$ about an axis through its center and is then lowered to a horizontal surface and released, as shown in Problem 64 (see Fig. P11.64). Furthermore, assume that the coefficient of friction between the disk and the surface is $\mu$. (a) Show that the time it takes for pure rolling motion to occur is $R \omega_{i} / 3 \mu g .$ (b) Show that the distance the disk travels before pure rolling occurs is $R^{2} \omega_{i}^{2} / 18 \mu g$.

Key Concepts

Moment of Inertia

The moment of inertia for a disk, given by I = (1/2)MR², is a fundamental property that quantifies the disk’s resistance to changes in its rotational motion. It plays a key role in linking the frictional torque to the resulting angular acceleration and is instrumental in calculating the time and distance required to transition from slipping to pure rolling.

Newton’s Second Law for Rotation

In the rotational context, the friction force also creates a torque about the disk’s center, leading to an angular deceleration. The relationship between torque (? = fR), moment of inertia (I), and angular acceleration is given by ? = I?. For a disk, understanding this effect is essential in determining how quickly the disk’s rotational speed decreases until the rolling condition is satisfied.

Frictional Force and Kinetic Friction

Kinetic friction, characterized by the coefficient ?, acts to both accelerate the translational motion and decelerate the rotational motion of the disk during the slipping phase. This friction force is responsible for converting some of the initial rotational kinetic energy into translational kinetic energy until the pure rolling condition v = ?R is met.

Pure Rolling Condition

This is the state in which an object rolls without slipping, meaning its translational velocity (v) is exactly related to its angular velocity (?) by the relation v = ?R. This condition is crucial because it defines the point at which the sliding phase ends and rolling motion becomes smooth and sustained without energy losses due to slipping.

Newton’s Second Law for Translation

This principle provides the framework for understanding how the frictional force influences the translational acceleration of the disk’s center of mass. The frictional force (f = ?Mg) produces a constant acceleration (a = ?g) that helps the disk gain linear velocity while overcoming its initial state of pure rotation.

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