Center of mass of cone using cylindrical coordinates

• xdrgnh
In summary, you used the equation of the cone to calculate the radius of the top of the cone. You should have used the equation of the cone to calculate the radius of the top of the cone.

Homework Statement

Set up intergral expression for center of mass of a cone using cylindrical coordinates with a given height H and radius R

Homework Equations

rdrddθdz is part of the inter grand. M/V=D volume of cone is 1/3π(r^2)H

The Attempt at a Solution

dm=Kdv dv=drdθdx K is just a constant because it is uniform density and mass. ∫∫∫zdrdθdz
z=z. Equation for cone in cylindrical coordinates is Z=(H/R)r. The top is bounded by Z=H. The angle is from o to 2π and the radius is square root of H^2+R^2. Once I take this integral I then divide by the same integral but instead there is no extra z in the inter grand. After all is set in done I get the wrong answer. I believe my initial set up might be wrong. Can anyone verify that or give a helpful suggestion thanks.

Hard to follow but what did you use for your lower limit on the inside dz integral?

[Edit: Woops I misread your order of integration... wait a sec...Ok, if you integrate dr first, r goes from zero to the cone. What did you use for r on the cone?

for r I used the square root of H^2+R^2

xdrgnh said:
for r I used the square root of H^2+R^2

But that is a constant. The radius varies as z moves up the cone. Use the equation of the cone to get r in terms of z for your lower limit.

I used that for the dr for the dz I used H/R(r) as the lower limit.

xdrgnh said:
I used that for the dr for the dz I used H/R(r) as the lower limit.

Your integrals are showing r dr dθ dz in that order. In that order you would integrate z last and its limits must be constant. Order matters so what order are you using?

$$\int_{lower}^{upper}\int_{lower}^{upper}\int_{lower}^{upper} (...)rd?d?d?$$

Quote this in your reply and fill in between the brackets and the question marks with what you did. Then it will be easy to see what you did wrong.

LCKurtz said:
Your integrals are showing r dr dθ dz in that order. In that order you would integrate z last and its limits must be constant. Order matters so what order are you using?

$$\int_{(H/R)r}^{H}\int_{0}^{2∏}\int_{0}^{R} (...)rd?d?d?$$

Quote this in your reply and fill in between the brackets and the question marks with what you did. Then it will be easy to see what you did wrong.
$$\int_{0}^{2∏}\int_{0}^{R}\int_{(H/R)r}^{H} (...) zrdzdrdθ$$

xdrgnh said:
$$\int_{0}^{2∏}\int_{0}^{R}\int_{(H/R)r}^{H} zrdzdrdθ$$

OK. That integral is correct for your numerator when calculating $\bar z$. When you divide it by the volume you should get $\bar z$ providing you don't make any algebra mistakes. Good luck, I have to go now.

1. What is the formula for finding the center of mass of a cone using cylindrical coordinates?

The formula for finding the center of mass of a cone using cylindrical coordinates is (0, 0, h/4) where h is the height of the cone.

2. How do you convert the coordinates of a cone from cartesian to cylindrical?

To convert coordinates of a cone from cartesian to cylindrical, you can use the following formulas:
x = rcosθ
y = rsinθ
z = z

3. Can the center of mass of a cone be located outside of the cone?

No, the center of mass of a cone will always be located within the cone itself.

4. How does the shape of a cone affect its center of mass?

The shape of a cone does not affect its center of mass. The center of mass will always be located at a height of h/4, regardless of the shape of the cone.

5. Is it necessary to use cylindrical coordinates to find the center of mass of a cone?

No, it is not necessary to use cylindrical coordinates. The center of mass of a cone can also be found using cartesian coordinates or other coordinate systems, but cylindrical coordinates are often the most convenient for this particular problem.