How to Calculate the Center of Mass for a Rotational Body Using Integration?

[Pir]

The curve y = sinx\sqrt{3cosx}, <br /> 0\leq x\leq \pi /2 rotates around the x-axis and creates a homogenous rotational body K.

a) Decide the volume of K.

b) Decide the center of mass for K.

(The x-coordinate of the center of mass is X_{T} = \frac{1}{m}\int_{K}^{} x dm, where m is the mass of of K.)

I have decided a) and I got the volume to be ∏ volume units. I need help with b). I would need to have the physics of the problem explained.

Please help me explain the physics, so that I from it can understand how to solve b).

 

[SteamKing]

What don't you understand about center of mass?

 

[Pir]

I know how to get to the formula X_{T} = \frac{1}{m}\int_{K}^{} x dm although here it is already given in the task. What I need help with is how to use it. First of all I wonder if - and if so, why - you are supposed to assume that the volume is varying inside the body, when you've already have calculated the total volume of body? The densitity is homogenous and can be set to equal 1, so we don't have to think about that. The parameter that can affect the center of mass is therefore the mass. Then I wonder, WHY does the mass in a body vary if the density is homogenous?

Lot of questions here, please answer as much as possible, thanks! :)

 

[SteamKing]

The mass is not varying inside the body, but the moment of a portion of the mass is.
The idea behind the center of mass is to find the point at which the entire mass would act if the body were under the influence of say gravity. In other words, the center of mass would represent a point where the mass would be balanced.

The location of this point is calculated by determining the moment of a small piece of the mass (dm) about a fixed location. In this instance, the fixed location is the y-axis. Therefore, the moment of dm becomes x*dm. To find the total moment, one must integrate x*dm. The x-coordinate will then be the total moment / total mass.

In the problem above, the density is constant and equal to 1. For bodies which are not homogenous, where the density can vary as a function of position, the dm in the integral would be expressed as the density multiplied by an element of volume, dV.

 

[Pir]

Aha, the moment varies throughout the body, not the mass, that makes sense.

So the formula for XT will be

X_{T} = \frac{1}{m}\int_{K}^{} 3\pi x sin^2xcosx dx

How do we calculate this integral? m is unknown.

 

[SteamKing]

m was calculated, in part, earlier in the problem. Remember, mass is density times volume. If you have calculated volume, then mass = rho * volume, where rho is the density.

 

[Pir]

So since rho = 1, mass is here the same as the volume. So then we get:

X_{T} = \frac{1}{\pi}\int_{K}^{} 3\pi x sin^2xcosx dx

Is this correct?

 

[SteamKing]

Looks good.

 

[Pir]

Are you sure? 1/pi and pi will cancel out the pi. How do I calculate this then?

And btw, could please not answer so short, I have a test in two days and I need to know this by then and it takes extremely long to solve just one task if I have to wait for the answer on every step. Please write more thorough solution (if you know it?)