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

[Hoppa]

any help on this question would be very appreciated :) thanks

A body in the shape of a cone has a circular base of radious R and a height h, from the centree of the base to its tip. The body is of uniform density p. Calculate the following characteristics of the cone:

A) its mass
b) the position of its centre of mass
c) its moment of inertia matrix, evaluated with respect to the centre of mass

 

[vincentchan]

Part a and b need a simple integral... choosing a right coordinate system will save you lot of time
For part c, if you choose your axis carefully, you will get a diaganol matrix and you only need to do two integrals...

 

[Hoppa]

sorry, what simple integral? using what?, I am pretty weak with this type of maths

 

[vincentchan]

A volume integral, or 3 dimensional integral...
If you have no knowlegde about 3 D integral, part c of the problem might be too advance for you...

 

[Hoppa]

yeh i don't know what a 3 dimensional intergral is. but for parts a and b then, what intergral do i use? what values do i use in the intergration? h1 and h2?

 

[vincentchan]

This link might help
http://www.mph.net/coelsner/calcapps/cone_ex.htm
what you need to do is basically find the volume of the cone and multiply by its density to get its mass...

 

[Hoppa]

thanks for that link. it really helped. think i got a basic understanding of it now. will try to work on part c as well

 

[Hoppa]

How do i find the position of the centre of mass? like what do i use for that?

 

[OlderDan]

I'm assuming you used the approach illustrated at the link to do the volume integral. The center of mass is an integral over the same limits. Instead of integrating dV = Adx to find volume, you need to integrate xdm where dm is a differential bit of mass and divide by the total mass. dm is density*dV, so what you need to do is integrate Axdx instead of Adx. This new integral is very similar to the one you did to find the volume. Make sure you get the constants (some combination of density, total mass, total volume) in the right places. The result should be a distance in this case, not a volume.

 

[Hoppa]

differential bit and divide by the total mass? what does that mean? i got the total mass from part a, but what's the differential bit? that i need to divide by total mass?

 

[vincentchan]

$$\int^h_x Axdx = \int^x_0 Ax dx$$
solve for the x...
h is the height of the cone, and A is the cross section area of the cone...

 

[OlderDan]

differential bit and divide by the total mass? what does that mean? i got the total mass from part a, but what's the differential bit? that i need to divide by total mass?

In the diagram, the plane that you can move up and down has the area A of a circle of radius s and a thickness dx, so it has a volume dV = A*dx. If the density of the material is D, the mass of this slice of the material is D*dV, and we call this a differential mass, dm. By definition, the center of mass is the sum of all the little masses multiplied by the coordinate position of that mass divided by the total mass of the object. It is really a 3 dimensional thing, but by symmetry we know the center of mass is on the x axis, so we only worry about the x coordinate. By substitution

x*dm/M = x*D*dV/M = x*D*A*dx/M = (D/M)*A*x*dx

D/M is a constant, but A is a function of x as shown in the diagram. Apart from the extra constants, the integral you have to do is identical the integral you did to find the mass except for the additional factor of x. So instead if integrating x^2, you will wind up integrating x^3.

It turns out that an equivalent way to define the center of mass is that it is the position that divides the integral into tow equal parts as noted by vincentchan. I prefer to change the symbol for the integration limit to something different from the integration variable to avoid any possible confusion.

$$\int^h_c Axdx = \int^c_0 Ax dx$$

where c is the x-coordinate of the center of mass.

 

[Hoppa]

ok i am still confused with this centre of mass problem. in my textbook, and then in my study guide i have got two other equations to find it :( i don't know how to implement any of them

 

[OlderDan]

ok i am still confused with this centre of mass problem. in my textbook, and then in my study guide i have got two other equations to find it :( i don't know how to implement any of them

If you will write a brief explanation of how you worked out the mass of the cone, I will use that as a starting point to tell you how to do the center of mass. So far it seems you have not been able to recognize the connection, so I think you need to review the mass calculation (volume calculation) before moving on.