91Ó°ÊÓ

Linear speed, or linear velocity, refers to how fast a point on a moving object travels along a straight path. In this context, we are analyzing a point on a vacuum cleaner belt.
The key formula connecting linear speed with rotational motion is \[ v = r \omega \] where:

• \( v \) is the linear speed of the point,
• \( r \) represents the radius of the rotating object, and
• \( \omega \) is the angular velocity.

This formula shows how the rotation of a shaft or wheel gets translated into linear motion.
When the shaft turns, it drags the belt along, resulting in a linear speed equal to the speed at the shaft edge. This speed remains constant around the wheel since there's no slip. Calculating it involves converting angular speed from revolutions per second to radians per second, then multiplying by the shaft’s radius. This gives the speed at which the belt moves linearly over time.

Angular velocity quantifies how quickly an object rotates around an axis, expressed in radians per second (rad/s).
For rotating objects like wheels, angular velocity is crucial in linking the object's rotation to real-world effects, like driving a belt.
In the given problem, we find the wheel's angular velocity using the constant speed of the belt (since it doesn’t slip) and the formula \[ v_w = r_w \omega_w \].Given the wheel's radius, this formula can be rearranged to solve for \( \omega_w \), which gives us how fast the wheel spins in terms of its rotation angle per second. By knowing the wheel's angular velocity, we see how quickly it contributes to the roller's rotation for specific functions like beating dirt from a rug.
Angular velocity helps bridge the gap between linear movement and rotational motion, illustrating how different parts of a machine interact.

Moment of inertia is a key physical concept in understanding rotational motion, yet it wasn't directly calculated in this exercise. Consider it as the rotational counterpart to mass in linear motion.
It determines an object's resistance to changes in its angular motion. Each part of a rotating system contributes to the total moment of inertia based on its mass and distance from the rotation axis.
While not part of the specific calculations for belt speed or wheel angular velocity, understanding the moment of inertia can deepen insights into how machines like vacuum cleaners operate. It influences how energy is distributed through rotating parts, affecting the machine’s efficiency and performance.
Even though we did not calculate it here, recognizing its significance helps in grasping the broader context of rotational motion in mechanical systems. It ensures balance in the entire mechanism, contributing to its effective and smooth operation.