- #1

KEØM

- 68

- 0

## 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

[itex]V \sim l^{3}[/itex] and that the surface area scales as [itex]S \sim l^{2}[/itex].

**Part B**: Show that this implies that [itex]S \sim V^{\frac{2}{3}}[/itex]

## Homework Equations

[itex]x = [L], y = [L], z = [L][/itex]

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.

[itex]V = (x)(y)(z) \sim [L][L][L] = [L]^{3}[/itex]

The same can be done for the surface area instead we will just use the length and width of the object:

[itex]S = (x)(y) \sim [L][L] = [L]^{2}[/itex]

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:

[itex]SV^{\alpha}\sim [1][/itex]

[itex]l^{2}[l^{3\alpha}]\sim [1][/itex]

[itex]\Rightarrow 2 + 3\alpha = 0 \Rightarrow \alpha = -\frac{2}{3}[/itex]

Plugging α back into the equation above gives us:

[itex]SV^{-\frac{2}{3}}\sim [1][/itex]

This then implies that:

[itex]S=V^{\frac{2}{3}}[/itex]

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.

Thanks in advance.