Volume of Sphere: Find w/ Pappus Theorem

[Jbreezy]

Homework Statement

Use the theorem of pappus to find the volume of the given solid
A sphere of radius r

Homework Equations

V = 2∏xA

The Attempt at a Solution

V = 2∏(4r/3∏)(4∏r^2) = (16/3)∏r^3

So something is wrong I should end up with (4/3)∏r^3 no?

 

[Dick]

Homework Statement

Use the theorem of pappus to find the volume of the given solid
A sphere of radius r

Homework Equations

V = 2∏xA

The Attempt at a Solution

V = 2∏(4r/3∏)(4∏r^2) = (16/3)∏r^3

So something is wrong I should end up with (4/3)∏r^3 no?

I don't think this makes any sense at all, which may be why you got the wrong answer. Which theorem of Pappus are you using? What surface are you revolving to get the sphere, and where is it's centroid?

 

[Jbreezy]

I used the surface area of a sphere. 4PIr^2 and then the (4r/3)PI is centroid of semi circle from the example they told me to use in my book.

Should I use a circle ? V = 2PI(4r/3PI)(PIr^2) still doesn't make sense though.

 

[SteamKing]

What's the area of a semicircle of radius R?

 

[Dick]

I used the surface area of a sphere. 4PIr^2 and then the (4r/3)PI is centroid of semi circle from the example they told me to use in my book.

Should I use a circle ? V = 2PI(4r/3PI)(PIr^2) still doesn't make sense though.

No, still doesn't. You still haven't stated coherently what you are doing. What theorem of Pappus and what are the parts?

 

[Jbreezy]

Theorem of Pappus


Let R be a plane region that lies entirely on one side of a line l in the plane. If R is rotated about l, then the volume of the resulting solid is the product of the area A of R and the distance d travled by the centroid of R.

Question: Use the theorem of Pappus to find the Volume of the given solid.

44.) A sphere of radius r (use example 4)
Example 4 is Find the center of mass of a semicircular plate of radius r.

The result is: The center of mass is located at the point 90, 4r/(3PI))


OK so the question I'm owrking in is in bold.

Can you help me now is this enough information? Should I not end up with the volume of a sphere V = (4/3)∏r^3 ?

 

[ArcanaNoir]

What are you using for the area of the semicircle?
The plane you are rotating is a semicircle, so you need to use its area (the area of a semicircle).

 

[Jbreezy]

Should I use (1/2)PIr^2 ?

 

[ArcanaNoir]

Yes :) That would be the area of a semicircle, and should give you the right result.

 

[SteamKing]

If jbreezy would check his work and the suggestions in posts #4 and #7, the answer would fall out immediately.

 

[Jbreezy]

I'm sorry this doesn't make any sense to me. So I have the area of the semi circle that is A in the equation
V = 2PIxA where x is the distance travled by the centroid? So I would have V = 2PI(PIr^2)((1/2)PIr^2 )
I used PIr^2 for x.

 

[Dick]

I'm sorry this doesn't make any sense to me. So I have the area of the semi circle that is A in the equation
V = 2PIxA where x is the distance travled by the centroid? So I would have V = 2PI(PIr^2)((1/2)PIr^2 )
I used PIr^2 for x.

You are garbling the theorem. The theorem says V=Ad, where d is the distance traveled by the centroid. Since the centroid travels in a circle if the radius of that circle is x then the distance d=2*pi*x. So x is the RADIUS of the circle the centroid makes, not the distance traveled by the centroid. So what is x?

 

[Jbreezy]

You are garbling the theorem. The theorem says V=Ad, where d is the distance traveled by the centroid. Since the centroid travels in a circle if the radius of that circle is x then the distance d=2*pi*x. So x is the RADIUS of the circle the centroid makes, not the distance traveled by the centroid. So what is x?

I'm not garbling anything. I'm giving you the exact words from my book they are the ones who write this trash.

I did this

V = Ad = 2PI(4r/3PI)(PIr^2/2)

A = (PIr^2/2)
d = 2PI((4r/3PI))
Is this right?

 

[Dick]

I'm not garbling anything. I'm giving you the exact words from my book they are the ones who write this trash.

I did this

V = Ad = 2PI(4r/3PI)(PIr^2/2)

A = (PIr^2/2)
d = 2PI((4r/3PI))
Is this right?

Yes, but you should write the centroid radius more carefully to make it clear the pi is in the denominator. It's 4r/(3pi). Doesn't V come out to be what you expect?