# Find the center of mass of the solid

• richies
In summary, to find the center of mass of a solid figure similar to a cone pointing upward with slope 1 and varying density, use integration to determine the mass and components of each disk at different heights. The radius of each disk is given by (h-z) and the overall mass can be represented as an integral from 0 to h of pi(h-z)^2*z^2dz. The center of mass in the z direction can then be found by taking the weighted sum of the mass elements divided by the overall mass.
richies

## Homework Statement

Find the center of mass of the solid figure similar to a cone pointing upward with slope = 1
Note: the density varies with z^2 and the edge has a slope of 1. From symmetry we see that both Xc and Yc are equal to zero. Find the center of mass in the z direction as a function of h by doing the appropriate integral.

## Homework Equations

p(vector r) = z^2 z^
slope = 1

## The Attempt at a Solution

I'm thinking about using cylindrical coordinates

@ radius = sqrt of (1 - h^2)

@ Z = 1/V ∫ z dV (lower limit V)

@ V = 1/3*pi*r^2*h

@ dV = r⊥dr⊥dθ dz.

@ ∫ z dV (lower limit V) = ∫ (∫ (∫zr⊥dr⊥) dθ) dz

Now I am stuck there i don't know if I am doing it right or wrong.
Any help or idea ?

#### Attachments

• geo_cone2.gif
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If the sides are straight and with slope 1, wouldn't the radius at height z be (h - z) ?

Since the density varies with height you'll need to find the mass of the object via an integration; you can't just use the volume of the object as a stand-in for mass.

"wouldn't the radius at height z be (h - z) ?"
is that the radius at the base depends on the change in height ? and it is equal h - z ?
if so, would the integral R = 1/M ∫p(r)rdV solve this problem?

richies said:
"wouldn't the radius at height z be (h - z) ?"
is that the radius at the base depends on the change in height ? and it is equal h - z ?
if so, would the integral R = 1/M ∫p(r)rdV solve this problem?

The radius at the base is given by (h - z) when z=0. That is, the radius at the base is h. As z increases the radius grows smaller. When z=h you've reached the apex of the object and the radius is zero there.

To find the center of mass you want the weighted sum of mass elements, dm, as you go up the z-axis, divided by the overall mass of the object. You need to determine an expression for an appropriate dm.

So my interation will be like this, correct me if I am wrong
r⊥ = h-z
r sqr(z^2+(h-z)^2)
M = integral from 0 to h of pi(h-z)^2*z^2dz
*
components:
x=0
y=0
z = (1/M)*∫ (from 0->h) ∫(from0->2pi)∫(from 0->h)(sqr(z^2+(h-Z)^2))drdθdz.

richies said:
So my interation will be like this, correct me if I am wrong
r⊥ = h-z
r sqr(z^2+(h-z)^2) <----
What does this represent?
M = integral from 0 to h of pi(h-z)^2*z^2dz
Okay, so the overall mass is the sum of the dm's, which are individually disks of radius (h-z) and thickness dz with density z2. That looks okay (although technically you're told that density varies as z2, so you should write ρ = k*z2).
components:
x=0
y=0
z = (1/M)*∫ (from 0->h) ∫(from0->2pi)∫(from 0->h)(sqr(z^2+(h-Z)^2))drdθdz.
You really only need a single integral over the dm's if you take each dm as a disk; you know that the center of mass of a disk is at its center and you know the mass of each disk via its radius and density.

thank you so much, now I get it :D

## 1. What is the definition of center of mass?

The center of mass is the point in an object or system where the entire mass can be considered to be concentrated. It is the point at which the object or system can be balanced.

## 2. How is the center of mass calculated for a solid object?

The center of mass for a solid object can be calculated by finding the average position of all the individual particles that make up the object. This can be done using the formula: center of mass = (sum of all individual particle positions) / (total mass of the object).

## 3. What factors affect the center of mass of a solid?

The center of mass of a solid can be affected by the shape, size, and distribution of mass within the object. For example, a larger mass on one side of the object will shift the center of mass towards that side.

## 4. Why is finding the center of mass important in physics?

The center of mass is important in physics because it helps us understand how an object will behave when it is subjected to external forces. It is a key concept in determining the stability, equilibrium, and motion of an object.

## 5. Can the center of mass be outside of the object?

Yes, the center of mass can be outside of the object if the distribution of mass is not symmetrical. In this case, the object will rotate around its center of mass when it is subjected to external forces.

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