Скачать или смотреть The relationship between linear and angular velocity: derivation of v=r*omega for circular motion.

Second video in a three-part series on uniform circular motion kinematics:

1. Angular speed and angle as a function of time for uniform circular motion:    • Angular speed and angle as a function of t...  

2. The relationship between linear and angular velocity:    • The relationship between linear and angula...  

3. Equations of motion for an object in uniform circular motion + derive the centripetal acceleration formula v^2/r:    • Equations of motion for uniform circular m...  

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00:00 We begin with an animation of a uniformly rotating disk that has a spot marked on it. The spot is an object in uniform circular motion! We allow a small increment of time delta(t) to elapse as the spot traces out a small increment of arc. We call the subtended angle delta(theta), the radius of the curve, r, and the arc delta(s).

00:27 Applying the arc length formula s=r*theta, we can now relate the arc length and angle as delta(s)=r*delta(theta).

00:44 Next, we divide by the time it took to trace the arc: delta(t). This gives us delta(s)/delta(t)=r*delta(theta)/delta(t). The left hand side of this equation tells us the distance traveled by the spot over the time it took, in other words that's the speed of the spot, and we use a "v" for the tangential speed of the object. On the right hand side, we recognize delta(theta)/delta(t) as the angular velocity for the rotation, which we call omega. Finally, we arrive at the formula relating linear and angular velocity for circular motion: v=r*omega!