# Exploring Flying Objects: Volume & Surface Area

• KEØM
In summary, the Buckingham Pi theorem states that the surface area of an object scales as the length cubed, while the volume scales as the length squared. This relation can be expressed using the equation SV^-\frac{2}{3} which implies that the surface area of an object is equal to the product of its length and width. Additionally, the volume of an object is equal to the product of its length, width, and height.
KEØM

## Homework Statement

Suppose we consider different flying objects, and that each object is characterized by a linear dimension l.

Part A: Use dimensional arguments to show that the volume V scales with size as
$V \sim l^{3}$ and that the surface area scales as $S \sim l^{2}$.

Part B: Show that this implies that $S \sim V^{\frac{2}{3}}$

## Homework Equations

$x = [L], y = [L], z = [L]$

where x, y, and z are the length, width and height of the object respectivley.

## The Attempt at a Solution

The volume of the object will be equal to the product of its length, width and height which all have the dimensions of length. Finding the product of these three will provide us with the scaling relationship.
$V = (x)(y)(z) \sim [L][L][L] = [L]^{3}$

The same can be done for the surface area instead we will just use the length and width of the object:
$S = (x)(y) \sim [L][L] = [L]^{2}$

I am not sure if this is how the problem is asking me to do the problem using dimensional analysis and scaling relations. Can anybody please verify for me if this is correct? It seems really basic (almost too basic) and I want to use the correct method to solve the problem.

For part b, I use the Buckingham Pi theorem as shown below:

$SV^{\alpha}\sim [1]$
$l^{2}[l^{3\alpha}]\sim [1]$
$\Rightarrow 2 + 3\alpha = 0 \Rightarrow \alpha = -\frac{2}{3}$
Plugging α back into the equation above gives us:
$SV^{-\frac{2}{3}}\sim [1]$
This then implies that:
$S=V^{\frac{2}{3}}$

Can someone pleae verify this for me? I feel like I should inlclude a dimensionless constant [tex]C[/itex] in my answer but I am not sure.

KEØM said:

## Homework Statement

Suppose we consider different flying objects, and that each object is characterized by a linear dimension l.

Part A: Use dimensional arguments to show that the volume V scales with size as
$V \sim l^{3}$ and that the surface area scales as $S \sim l^{2}$.

Part B: Show that this implies that $S \sim V^{\frac{2}{3}}$

## Homework Equations

$x = [L], y = [L], z = [L]$ where x, y, and z are the length, width and height of the object respectivley.

## The Attempt at a Solution

The volume of the object will be equal to the product of its length, width and height which all have the dimensions of length.
This is, strictly speaking, only true for a rectangular solid- and you don't know the shape of these objects (or even if they have the same shape). Of course, for any shape, the volume is a dimensionless constant times the product of three lengths, not necessarily "length, width, and height".

Finding the product of these three will provide us with the scaling relationship.
$V = (x)(y)(z) \sim [L][L][L] = [L]^{3}$

The same can be done for the surface area instead we will just use the length and width of the object:
$S = (x)(y) \sim [L][L] = [L]^{2}$
Again, "area equals length times width" is true only for a rectangle. What is generally true is that the area of a figure is some dimensionless constant times the product of two lengths, not necessarily "length and width".

I am not sure if this is how the problem is asking me to do the problem using dimensional analysis and scaling relations.
Well, that is what the problem said, isn't it? "Use dimensional arguments".

Can anybody please verify for me if this is correct? It seems really basic (almost too basic)
Not "almost". Assuming a rectangular solid is too basic.

and I want to use the correct method to solve the problem.

For part b, I use the Buckingham Pi theorem as shown below:

$SV^{\alpha}\sim [1]$
$l^{2}[l^{3\alpha}]\sim [1]$
$\Rightarrow 2 + 3\alpha = 0 \Rightarrow \alpha = -\frac{2}{3}$
Plugging α back into the equation above gives us:
$SV^{-\frac{2}{3}}\sim [1]$
This then implies that:
$S=V^{\frac{2}{3}}$

Can someone pleae verify this for me? I feel like I should inlclude a dimensionless constant [tex]C[/itex] in my answer but I am not sure.

So to better describe the volume of any object could be better described using the following equation:

$V=C(l_{1})(l_{2})(l_{3})$ , where C is a dimensionless constant and $l_{1} l_{2}, l_{3}$ are the lengths of the object.

In the same manner we can write the formula for the surface area as shown below.

$S=C(l_{1})(l_{2})$ where C once again is a dimensionless constant and $l_{1} l_{2}$ are the lengths of the object.

Knowing that the dimensions of $l_{1}, l_{2}, l_{3}$ are all length $[L]$ I can use the same process as I did originally.

Then we can say that independent of what the dimensionless constant C is the volume will scale with $[L^{3}]$ and the surface area will scale with $[L^{2}]$.

I now realize that I should change my final result to:

$S=C(V^{\frac{2}{3}}) \Rightarrow S \sim V^{\frac{2}{3}}$

Is this now correct? Did I use the Buckingham Pi theorem correctly in order to solve this problem?

Thanks again.

## 1. What is the difference between volume and surface area?

Volume refers to the amount of space occupied by a three-dimensional object, while surface area refers to the total area of the object's outer surface.

## 2. Why is it important to calculate the volume and surface area of flying objects?

Calculating the volume and surface area of flying objects can help scientists understand the object's physical properties and how it interacts with the air. This information is crucial for designing and improving aircrafts and other flying vehicles.

## 3. How do you calculate the volume of a flying object?

The volume of a flying object can be calculated by multiplying its length, width, and height. For more complex shapes, such as airplanes, the volume can be calculated by dividing the object into simpler shapes and using their respective volume formulas.

## 4. Can the surface area of a flying object affect its flight?

Yes, the surface area of a flying object can affect its flight by increasing or decreasing the amount of air resistance the object experiences. Objects with a larger surface area will experience more air resistance, which can impact their speed, stability, and fuel efficiency.

## 5. How does the volume and surface area of a flying object change as it flies?

The volume and surface area of a flying object remain constant as it flies, unless the object changes shape or size. However, the object's orientation and position can affect the amount of air resistance it experiences, which can indirectly impact its volume and surface area.

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