Center of Mass of a Cone?

Question

I am still not understanding it: Here is the question

Find the ' y ' component of the center of mass for a uniform solid cone of radius ' R ' and height ' h '. The answer should be expressed as a (numeric) fraction of ' h ', measured from the base of the cone.

I found an example that worked out how to find the center of mass of a square pyramid.

The way they worked it out as follows, they said that the mass of a slab of thickness dy at a height y is

dm = p(Density) * dV.

Then they said the Y coordinate of center of mass is
Ycm = 1 / M * S(Integral) y dm

They then said that the total mass is

M = S(Integral) dm = S(Integral) slab thickness dy

They then got Ycm = S(Integral) y^3 dy / S(Integral) y^2 dy

They then took that expression and integrated from 0 to h, and expressed Ycm as a fraction of h. I don't really understand this and how to apply it to a cone. A thorough step by step explanation in words along with the mathmatical expressions will really help.

Answer 1

It sounds like you're having trouble understanding the sensibility of the definition of "center of mass" in integral form. It's not an easy thing to understand on first blush.

Sometimes it helps to think about the point-mass form of the definition first. A collection of point masses Mi, each of which has some nominal displacement Yi from a defined origin has the following center of mass:

Yc = sum(MiYi) / sum(Mi)

Try a few simple cases to convince yourself that this is true.

Similarly, there is an integral form of center of mass for a continuum of mass elements dM. In this case, the sums convert to integrals, and

Yc = Int{Yi*dM} / Int{dM}

Now let's apply this case to your right-circular cone. Since you're looking for the vertical center of mass, divide your cone into horizontal disks of mass dM = p*dV.

It pays to recognize that p (density) is constant. That means you can pull it out of the integral ratio completely, and

Xc = Int{Xi*dV} / Int{dV} (This will save a step later)

Now in this case, dV = pi*r^2 dy, where r is the radius of the cone at an arbitrary height y, and dy is the thin vertical slice of your disk. The radius is dependent on height y, and because you're integrating over dy you need to express r in terms of y. At y=0, r=R, and y=h r=0. The progression is linear, meaning a point-slope representation of r as a function of y should be simple; I get r=R-yR/h between y=0 and y=h.

Re-expressing, dV = pi*R^2(1-y/h)^2dy

Now let's play:

Yc = Int{y*dV} / Int{dV}

Numerator first:

Int{y*dV} = Int[0,h] {pi*R^2(1-y/h)^2 y} dy
= pi*R^2*Int[0,h] {y-2y^2/h+y^3/h^2}dy
= pi*R^2*{y^2/2 - 2y^3/(3h) +y^4/(4h^2)} [0,h]
= pi*R^2*{h^2/2-2h^2/3+h^2/4}
= (1/12)*pi*R^2*h^2

Denominator next:

Int{dV} = Int[0,h]{pi*R^2(1-y/h)^2} dy
= pi*R^2{-h/3(1-y/h)^3}[0,h]
= (1/3)pi*R^2h (we knew this back in fifth grade, but it never hurts to check)

So Yc = {(1/12)pi*R^2*h^2} / {(1/3)pi*R^2*h}
Yc = h/4

I think that gives you everything you need. It's easier to talk through with a diagram, but Yahoo! isn't there yet.

Good luck, work hard, and stay away from drugs.

Answer 2

Center Of Mass Of Cone

Answer 3

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RE:
Center of Mass of a Cone?
I am still not understanding it: Here is the question

Find the ' y ' component of the center of mass for a uniform solid cone of radius ' R ' and height ' h '. The answer should be expressed as a (numeric) fraction of ' h ', measured from the base of the...

Answer 4

The mass of a slab of the cone at height y is rho*dy*π*r^2, assuming the density rho is a constant. The section of the cone at any height is a circle, and the mass of a circular disc of height dy is the volume of that disc (π*r^2*dy) times the densty. r is the radius of the cone at height y, and is a linear function of y. At y = 0, r = R, and at y = h, r = 0 therefore

r(y) = R- (R/h)*y

The mass is ∫rho*π*r(y)^2dy [ y = 0 to h]

So your integral is

∫rho*π*[R-(R/h)*y]^2*y*dy / ∫rho*π*[R -(R/h)*y]^2*dy

You can cancel the rho, R and π terms to get

∫[1 - (1/h)*y]^2*y*dy / ∫[1 - (1/h)*y]^2*dy

Answer 5

The answer is y = h/4, where h = the height of the cone.

I'm afraid that you will have to work out the detailed derivation, however.

Answer 6

hint:volume of cone=1/3 (volume of cylinder)