Question

Find the center of mass of a cylinder of radius 2 and height 4 and mass density e^-z, where z is the height above the base. (Please Will Give Medal and Follow!)

Answer 1

thank you!!!

Answer 2

If you put it on the X-Y Origin, on its circular base, by symmetry, where does this put the center of mass in the X and Y directions?

Answer 3

x - -2 to 2
y - 0 to 4 ?

Answer 4

i mean z 0 to 4

Answer 5

im a bit confused for the y

Answer 6

Hmmm. We are not yet breaking out any integral. The height of the cylinder has nothing to do with the Center of Mass in either the X or Y direction. It's just a circle sitting on the Origin. Where is its center?

Answer 7

(2,2) ?

Answer 8

Why would it be out there? That's not even on the circle. Now, where is that center?

Answer 9

oh okay wow yeah i see that (0,0)

Answer 10

so that means both the x and y the double integrals right? would go from -2 to 2 ? how would i set up the integral? thank u!

Answer 11

You can do that if you want, but it is entirely unnecessary. Think about the density function. It is a function of Z only. In other words, cutting the cylinder into really thin slices parallel to the XY-Plane, we see that each slice has uniform density. Do you believe?

Answer 12

yeah I do believe I think I got it hopefully thank you!

Answer 13

unless dan815 has anything to add?

Answer 14

This tells us that the center of mass is at (0,0,Something). All we need is the average value of Z. What integral did you use to produce what value?

Answer 15

integral -2 to 2 integral -2 to 2 and the function dx dy

Answer 16

You've a function, \(Mass = e^{-z}\). You need the average value for \(z\in [0,4]\). That's the height, right? How do we find the average value?

Answer 17

* Density, not Mass. Sorry about that.

Answer 18

1/V integral (3x) f(x,y,z) dV

Answer 19

i meant like the integra three times i know bad notation (wont use for exam) lol

Answer 20

We know that \(f(x,y,z) = e^{-z}\). What is 3x?

Answer 21

so all i need is volume then correct? how would i go about with the bounds? x: -2 to 2 y: -2 to 2 z: ?

Answer 22

thank you so much for helping me really means a lot

Answer 23

You seem to be missing the significance of the symmetry of the situation. In this case, you do NOT need the volume. No need to compute more than one integral. We already know (0,0,Something) is the center of mass. Don't calculate anything we already know. Since we already know the values we need for x and y, don't do ANY integral to find them. We already know them! We just need to find z. Since the average value of e^-z has nothing to do with x or y, it takes only one small integral. Calculate the average value of e^-z and you are done.

Note: it appears I previously stated that we need the average values of z. This was incorrect. We need the average value of e^-z.