Suppose you have a spool of thread that can rotate around a fixed axis as you pull thread off it. If you pull the thread with your hand so that the thread always lies along a certain fixed line, how will your hand's speed as it moves compare to $|\vec{\omega}|$, where $R$ is the spool's radius and $R|\vec{\omega}|$ is angular speed?

Question

Quizplus · Accepted Answer

The hand's speed will be equal to $R|\vec{\omega}|$ because the linear speed of the thread being pulled off the spool is directly related to the angular speed of the spool by the equation $v = R\omega$, where $v$ is the linear speed of the thread (or the hand's speed in this context), $R$ is the radius of the spool, and $\omega$ is the angular speed of the spool.

Suppose You Have a Spool of Thread That Can Rotate ∣ω⃗∣|\vec{\omega}|∣ω∣

Suppose You Have a Spool of Thread That Can Rotate $|\vec{\omega}|$