Connecting Linear and Rotational Motion
Topic 5.1 promised one thing: every point on a rigid rotating body shares the same $\omega$. Topic 5.2 is the consequence. The linear quantities at each point are not shared. A point near the hub creeps. A point at the rim flies. Same disk, same $\omega$, different $v$.
The bridge is one line, used three times. Arc length at a point: $s = r\theta$. Tangential speed: $v = r\omega$. Tangential acceleration: $a_t = r\alpha$. In each, the radius $r$ converts a shared angular quantity into a point-specific linear one. Drop $r$ and every wheel, turntable, or fan-blade problem comes out wrong.
Students go wrong two ways in 5.2. The first is dropping the factor of $r$: writing $v = \omega$ or $a_t = \alpha$ and treating two quantities with different units as the same number. The second is the “rim and hub share $v$” mistake. The rim and hub do share $\omega$; that part is right. But $v = r\omega$ makes $v$ scale with $r$, so different points on the same disk have different $v$. The skill check catches both before a test does.
The work
3 ways in · any order
Lesson
One Disk, Many Speeds
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A ten-scenario applet that drills the three linkages: arc length, tangential velocity, and tangential acceleration. Each scenario is one multiple-choice question; finishing all ten clears the topic.
Diagnostic
10-item topic check
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Ten multiple-choice items that test the two common errors: dropping the factor of $r$ from $v = r\omega$, and confusing $\omega$ (shared) with $v$ (not shared). Pass it and you can skip the lesson; miss items and the targeted-practice card lights up below.
Targeted Practice
Drill a single misconception
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Pick one of the failure modes you've missed and grind it on its own. The round is adaptive: two correct in a row clears the misconception and you move on.