A thin uniform rod is hinged at its midpoint to begin with...

A thin, uniform rod is hinged at its midpoint. To begin with, one-half of the rod is bent upward and is perpendicular to the other half. This bent object is rotating at an angular velocity of 6.5 rad/s about an axis that is perpendicular to the left end of the rod and parallel to the rod's upward half (see the drawing). Without the aid of external torques, the rod suddenly assumes its straight shape. What is the angular velocity of the straight rod? rad/s

Transcript

0:00 All right, hello.

00:01 In this question, we're given this setup, where we have a very thin rod, which is bent into an l shape.

00:08 And initially, it's like this.

00:10 So we have a thin half of the rod coming off to the left.

00:15 And then it bends and is spinning about an axis that is parallel to the other end of the rod.

00:22 So going up through the rod there.

00:25 And that is spinning about.

00:27 So think of it as kind of like a blade spinning about this hinge here.

00:33 And then we're told, without any external torques, the rod straightens out.

00:37 And so the only way for our, because it's a thin rod, that means that it's going to straighten out this way.

00:44 And then it will, again, continue to rotate about this axis, which is going straight up and down with some angular speed, omega final.

00:54 And we want to figure out what that omega final is.

00:56 So in order to do this, we're told that there's no external torques.

01:00 So we can use conservation of momentum.

01:02 That is the condition for conservation of momentum.

01:05 You don't have any external forces or torques.

01:07 So initially, we have some initial moment of inertia and some initial speed.

01:14 And then at the end, we have some final moment of inertia and some final angular speed.

01:18 Well, what is our moment of inertia initially? essentially, we can break this rod into two parts...

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