You are making a bowl out of a solid hemispherical piece of...

You are making a bowl out of a solid hemispherical piece of strange, dense wood (you may assume the center of the flat face is the origin). The wood has a density at each point that is proportional to the distance from the center of the flat face of the hemisphere. The initial piece of wood has a mass of 5/pi kg, and the center of mass in the z-direction is 2 inches below the flat face (into the wood). You then hollow out the wood to make your bowl so that the inside is the shape of a concentric sphere. (a) Find the radius of the original piece of wood (before hollowing it out), and the proportionality constant of the density.

Transcript

00:01 So we have a ball, the massive r, starting from rest, and it's moving down to the bottom, and we have to determine what the final angular velocity is.

00:15 So the first thing we want to do is find out.

00:18 We want to set initial equal to ef.

00:21 We're going to use conservation of energy.

00:23 That's k initial plus u initial is equal to k final plus u final.

00:31 So the potential energy will be completely converted to kinetic at the bottom.

00:35 And it starts with no kinetic energy.

00:39 So what we have is mgh is equal to 1�mv f squared plus one -half i omega -squared f -squared.

00:54 So omega -r is equal to v.

01:00 And the i in this case is of a uniform solid sphere, so it's 2 5th, mr squared.

01:06 So what we have is mgh is equal to 1 half, m times omega squared, r squared, plus 1 .5th times 2 .5th is 1 5th.

01:23 M times omega squared.

01:30 So we want to find out what is this height.

01:33 Okay? so what is this height? if we have a big r, then the circumference of this circle is the circumference is equal to 2 pi par.

01:48 Now we want the sliver of it.

01:51 So that means that we are going to, we're going to take this fraction of it.

02:13 So from here to here is going to be, we're going to call this a arc length.

02:31 The arc length is going to be.

02:36 So the equation for arc length is actually s is equal to.

02:46 We're going to call this s actually r theta so or in this case it's big r theta so it's going from here to here a distance of s so we want to find out how far away at this point from here in terms of height okay so it turns out we don't need the arc length at all to find out what this is because we can just use law of cosines so if we we basically have an isosceles triangle between this point and this point.

03:31 So if we have an isosceles triangle with side lengths of r and r, then we can use the theta value to find out what the distance between these two points is.

03:45 So that's going to be the law of cosines is a squared is equal to b squared plus c squared minus 2bc.

03:57 So a squared is equal to b squared plus c squared minus 2 minus 2bc cosine theta.

04:11 So that means that this distance a is equal to 2r squared minus 2r squared cosine theta, the square root of that.

04:37 So that's a is equal to 1 minus 2r squared theta times cosine, 2r squared times cosine theta.

05:01 So now we want to find what is the height based on that.

05:26 So we know the hypotenuse.

05:28 We don't know what this angle is.

05:32 But at this point, there's going to be a tangent.

05:36 So this is going to be, if this is an internal angle here, interacting with this tangent, tells me nothing.

05:52 I need to know what this angle is.

06:37 Okay, so if we have an angle data here, then this angle is going to be, so these two angles are given to us by 180 minus theta over 2.

06:54 So that means that this angle is 90 minus plus data 2.

07:09 So these two are going to cross off...

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