In Fig. 10-55, a wheel of radius 0.20 m is mounted on a frictionless horizontal axle. A massless

step 1 What is the rotational inertia? Our expression for our tension can be written as mass times g si

In Fig. 10-55, a wheel of radius 0.20 m is mounted on a...

In Fig. $10-55,$ a wheel of radius 0.20 $\mathrm{m}$ is mounted on a frictionless horizontal axle. A massless cord is wrapped around the wheel and attached to a 2.0 $\mathrm{kg}$ box that slides on a frictionless surface inclined at angle $\theta=20^{\circ}$ with the horizontal. The box accelerates down the surface at 2.0 $\mathrm{m} / \mathrm{s}^{2} .$ What is the rotational inertia of the wheel about the axle?

In Fig. $10-55,$ a wheel of radius 0.20 $\mathrm{m}$ is mounted on a frictionless horizontal axle. A massless cord
is wrapped around the wheel and attached to a 2.0 $\mathrm{kg}$ box that slides on
a frictionless surface inclined at angle $\theta=20^{\circ}$ with the horizontal. The
box accelerates down the surface at 2.0 $\mathrm{m} / \mathrm{s}^{2} .$ What is the rotational inertia of the wheel about the axle?

Key Concepts

Moment of Inertia

The moment of inertia measures an object's resistance to changes in its rotational motion. It depends on both the mass of an object and the distribution of that mass relative to the axis of rotation, serving as the rotational analogue to mass in translational dynamics.

Newton's Second Law for Rotation

This law states that the net external torque acting on a body is equal to the product of its moment of inertia and its angular acceleration (? = I?). It is fundamental for analyzing how forces produce rotational acceleration in a system.

Newton's Second Law for Translation

This principle, expressed as F = ma, relates the net force acting on an object to its mass and the resultant linear acceleration. It is essential for understanding the dynamics of systems where translational motion is influenced by external forces.

Relationship Between Linear and Angular Acceleration

In systems where rotation and translation are linked, such as a wheel and a connected mass, the linear acceleration of the mass and the angular acceleration of the wheel are connected by the wheel’s radius (a = ?R). This relationship allows the dynamics of one part of the system to be related to the other.

Torque and Tension

Torque is the rotational force produced when a force is applied at a distance from an axis. In systems involving a cord and a wheel, the tension in the cord generates a torque that results in angular acceleration. This relationship is crucial for linking the forces in translational motion to those in rotational motion.

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