How Do You Calculate the Center of Mass for a Cone?

[GreenPrint]

Homework Statement

Calculate the X_com of the cone of mass M in terms of quantities given in the picture. The density of the cone is uniform.

See the attachments for the picture.

Homework Equations

The Attempt at a Solution

When I did it I got

X_com = (3L)/2

and I am unsure if this is correct or not. The infinitely small sections are circular disks of area pi R^2 were r is the radius of the infinitely small disk and a volume just an infinitely small width dx times the area. therefore the density

rho = dm/dV = dm/(pi R^2 dx)

I sort of got confused when I did because the radius changes with respect to x and during the middle of the test I sort of rushed this problem and thought that it just canceled out in the end but believe I may be wrong

 

[SteamKing]

You have shown that the center of mass of the cone in the x-direction lies outside of the cone. Your answer is not correct.

Yes, the radius changes with respect to x, but it does so in a predictable manner.

Care to reformulate your solution to the c.o.m.? Concentrate on writing the moment equation using the origin as the reference point.

 

[GreenPrint]

what is the moment equation?

 

[SteamKing]

It's an essential component in your c.o.m. calculation.

d(moment)/dx = x * dm

 

[GreenPrint]

Im not exactly sure what is meant by moment in your equation. Can someone tell me. I recalculated it except this time I got 3/4 R. I believe the answer is 3/4 L though. I'm not exactly sure what I did wrong. Thanks for any help.

sorry that a four in the denominator magically disappeared from my work towards the end and magically reappeared. I just realized this. Everything else is correct though I believe, except some reason I got 3/4 R instead of 3/4 L