A solid disk of mass M = 0.600 kg and radius R = 0.250 m is...

A solid disk of mass M = 0.600 kg and radius R = 0.250 m is suspended vertically and is free to rotate about its center without friction. A point object that has the same mass as the disk is placed at an angle ?_i = 35.0° clockwise from the top along the rim, causing the disk to rotate. The acceleration due to gravity is g = 9.80 m/s^2. What is the angular speed ?_f of the disk when the object is directly below the center of the disk. ?_f = __________________ rad/s

Transcript

0:00 All right, hello.

00:01 In this question, we're given this setup where we have an object of mass m sitting on a wheel, solid wheel, of mass m.

00:10 And we're told it's released from us, so this object moves down.

00:14 And once it's at the bottom down here, it will be traveling, obviously, to the left and with some speed.

00:20 And we want to know when that happens, this whole wheel will be spinning, what is omega final going to be? we know that the omega 0 is equal to 0.

00:28 Well, in order to do this, we're going to use conservation of energy.

00:32 Seems like a good approach.

00:34 And so initially, what do we have? well, we have this mass, and it's elevated some distance h above where it's going to end up.

00:43 We also have mass stored in this disk.

00:48 But the center of mass of the disk doesn't actually change how high it is, so we don't have to worry about gravitational potential of that.

00:56 So we just have gravitational potential of the point mass, initially.

00:59 And at the end, we have the point mass, which is moving with a linear speed, 1 half mv.

01:06 So we have kinetic energy of the point mass.

01:09 It's not itself actually rotating at all, but the wheel is.

01:14 So we also have rotational kinetic energy of the wheel.

01:18 So let's go ahead and expand this out.

01:20 We're going to have m times g times h.

01:22 The particle is moving 1 half mv squared.

01:26 At the bottom, that's v final.

01:27 And then the object is rotating with 1 half i omega final squared.

01:33 Ok, well, we've got to clean this up a little bit.

01:36 So first off, we need to figure out what h is...

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