Dimensional analysis and frustum of a cone

In summary, the conversation is about finding the correct mensuration expression for the volume of a frustrum of a cone and using dimensional analysis to understand how the expression is equal to the volume. The conversation includes hints for solving the problem and a clarification of the expressions used. Finally, the person struggling with the problem answers their own question.
  • #1
selig
2
0

Homework Statement


Hi
Im having some difficulty with the following question:
Figure P1.14 shows a frustrum of a cone. Of the following mensuration (geometrical) expressions, which describes (a) the total circumference of the flat circular faces, (b) the volume, and (c) the area of the curved surface?
(i) π(r1 + r2)[h2 + (r1 – r2)2]1/2 (ii) 2π(r1 + r2) (iii) πh(r1^2 + r1r2 + r2^2)

and I am at part b.
Since I already know that the volume of a frustum of a cone is number (iii) I have to now prove it.
The problem is that I am having some difficulty showing it with the use of dimensional analysis.

Since V=[L^3]
how is possible that πh(r^12 + r1r2 + r2^2) is equal to it?
I know that r=L and h=L but completely confused on how to set it up
can someone point me in the right direction? I don't


thank you
 
Physics news on Phys.org
  • #2
First start off with a cone, say cone A. If you slice off a smaller cone, say cone B, from the "top" of cone A, the solid that remains is a frustrum.

For this question, I believe [tex]r_1[/tex] is the radius of the circular base of cone A (equivalently, the larger flat circular face of the frustrum) and [tex]r_2[/tex] is the radius of the circular base of cone B (equivalently, the smaller flat circular face of the frustrum). I think the quantity h refers to the height of the frustrum.

Here are some useful hints for solving part (b) of the question:
1) How do you find the volume of a cone?
2) Cones A and B are similar; use this to express the heights of cones A and B in terms of [tex]r_1[/tex], [tex]r_2[/tex] and h.
3) Note that [tex]x^3 - y^3 \ = \ (x-y)(x^2+xy+y^2)[/tex].

To clarify some of the expressions in your original post,
Option (i) [tex]\pi(r_1+r_2)\sqrt{h^2+(r_1-r_2)^2}[/tex]
Option (iii) [tex]\frac{1}{3}\pi h (r_1^2+r_{1}r_{2}+r_2^2)[/tex]
 
Last edited:
  • #3
thank you, I think I got it
 
  • #4
I'm struggling on practically this exact problem right now in my textbook.

The way I see it, I get this:

L (L2 + L2 + L2) = L3 + L3 + L3 = 3L3

But I don't see how I can arrive at L3 from my end answer...
 
  • #5
Meadman23 said:
I'm struggling on practically this exact problem right now in my textbook.

The way I see it, I get this:

L (L2 + L2 + L2) = L3 + L3 + L3 = 3L3

But I don't see how I can arrive at L3 from my end answer...

I just had to answer my own question here. The 3 disappears because it's dimensionless!
 

1. What is dimensional analysis?

Dimensional analysis is a mathematical technique used to convert units of measurement from one system to another. It involves breaking down a given quantity into its fundamental dimensions, such as length, mass, and time, and then using conversion factors to manipulate and cancel out units until the desired unit is achieved.

2. Why is dimensional analysis important in science?

Dimensional analysis is important in science because it allows for accurate and consistent measurements, regardless of the system of units used. It also helps to identify and correct errors in calculations, as well as understand the relationships between different physical quantities.

3. What is a frustum of a cone?

A frustum of a cone is a three-dimensional geometric shape that is formed by slicing a cone with a plane parallel to its base. It has two circular faces, a smaller one at the top and a larger one at the bottom, connected by a curved surface.

4. What are the applications of frustum of a cone?

The frustum of a cone has various applications in engineering and architecture. It is commonly used in the design of buildings, bridges, and tunnels, as well as in the construction of storage tanks and silos. It is also used in optics for the design of lenses and mirrors, and in manufacturing for the production of conical objects such as bottles and funnels.

5. How is the volume of a frustum of a cone calculated?

The volume of a frustum of a cone can be calculated using the formula V = (1/3)πh (R^2 + r^2 + Rr), where h is the height of the frustum, R is the radius of the larger circular face, and r is the radius of the smaller circular face. Alternatively, it can also be calculated by subtracting the volume of the smaller cone (πr^2h/3) from the volume of the larger cone (πR^2h/3).

Similar threads

  • Introductory Physics Homework Help
Replies
23
Views
1K
  • Introductory Physics Homework Help
Replies
4
Views
792
  • Introductory Physics Homework Help
Replies
5
Views
2K
  • Introductory Physics Homework Help
Replies
4
Views
924
  • Introductory Physics Homework Help
Replies
5
Views
1K
  • Introductory Physics Homework Help
Replies
2
Views
719
  • Introductory Physics Homework Help
Replies
3
Views
3K
  • Special and General Relativity
Replies
13
Views
1K
  • Introductory Physics Homework Help
Replies
2
Views
10K
  • Introductory Physics Homework Help
Replies
6
Views
3K
Back
Top