Physics 01

Introduction to Dimensions

Physics is concerned about the very small, and the very large. In order to represent our observations, we need to know about units. The SI unit for length is the metre (m), time - seconds (s) and mass - kilograms (kg). The L (length), T (time) and M (mass) are fundamental units which all other quantities in physics can be derived.

[ Speed ] = [ L ] / [ T]

The dimension of Speed, is the dimension of Length, divided by the dimension of Time.

[ Volume ] = [ L ]³

The dimension of Volume, is the dimension of Length to the power of 3.

[ Density ] = [ M ] / [ Volume ] = [ M ] / [ L ]³

[ Acceleration ] = [ L ] / [ T ]²

All other quantities can be derived from these 3 fundamentals.

Uncertainty

Any measurement, without any knowledge of its uncertainty, is completely meaningless. The uncertainty is the known error margin. Take for example, two measurements taken are:

25.0cm
24.5cm

Whether it is true that the subject in the first measurement is longer than the second one, or is there a possibility that they might be equal, depends on the error margin or the uncertainty of a measurement. If the uncertainty is +-0.1, we can almost guarantee that the first subject is longer than the second. However, if the uncertainty is +-1cm, we may have to take more measurements.

We can find the uncertainty of our measurements through control tests.

Galileo Proportions Problem

When it comes to measurements in systems, we also need to look at how certain measurements scale with each other. Let's look at an example posed by Galileo Galilei.

Suppose that we have an animal of size S. It has legs with a femur bone of length L. The femur bone has a thickness of D.

It is safe to say that an animal has a femur bone that is proportionate (∝) to its size:
S ∝ L

In this case, it is also safe to say that the mass of an animal is proportional to its size to the power of 3:
M ∝ S³

Therefore:
M ∝ L³

Now notice that in order to support its weight, according to Galileo, its femur bone's cross section must be proportionate to the mass:
M ∝ D²

We can then derive this proportionality:
D ∝ S^3/2

Then he raised a comparison between an elephant and a mouse. An elephant is about 100 times the size of the mouse, so its femur bone would be 100 times longer, which is true. This proportionality suggests that an elephant's femur cross section would also be 1000 times thicker than the mouse, which turned out to be false. The elephant's femur, is only 100 times thicker, which scientists concluded to be nature's protection against buckling.

Derivation of Dimensions

Let us look at how we can derive the dimensions of units. Recall the following formula:

F = ma

[ Force ] = [M] * [ Acceleration ]

[ Acceleration ] = [ Velocity ] / [ T ]

[Velocity ] = [ L ] / [ T ]

Therefore:

[ Acceleration ] = [ L ] / [ T ]²

[ Force ] = [ M ] [ L ] / [ T ]²

Now let's look at Momentum. Momentum is described as:

p = mv

[ Momentum ] = [ M ] * [ Velocity ]

Therefore:

[ Momentum ] = [ M ] * [ L ] / [ T ]

How about Pressure? Pressure is described as Force per unit Area:

[ Pressure ] = [ Force ] / [ Area ]

[ Force ] = [ M ] [ L ] / [ T ]²

[ Area ] = [ L ]²

Therefore:

[ Pressure ] = [ M ] / [ L ] [ T ]²

Finally, Kinetic Energy is defined as:

1/2*mv²

This gives us:

[ Kinetic Energy ] = 1/2 [ M ] [ L ]² / [ T ]²

From here, we can derive Work, which is the application of Force for a certain Distance:

[ Work ] = [ M ] [ L ]² / [ T ]²

We can also derive Power, which is the supply of Energy (which is equal to Work) over Time:

[ Power ] = [ M ] [ L ]² / [ T ]³

Period of Pendulum Problem

Let us now create a function that can tell us the period of a pendulum (time it takes for a full oscillation).

Suppose that we have the following information:
l = Length of Pendulum (L)
m = Mass of Bob (M)
g = Gravitational Acceleration (L/T²)
θ = Angular Amplitude (L)
C = Constant

Our final solution would be in the form:
[T] = C [L]^p [M]^q [L]^r [LT^-2]^s

From here, we can conclude that q is zero because it doesn't appear on the left at all. We can also conclude that s must be -0.5 because T on the left is to the power of 1 only. Since L doesn't appear on the left, then L should be eliminated as well.

The total power of L in this equation is currently p+r+s. In order for L to have the power of 0, we must have:
p+r+s = 0
p+r = -s
p+r = 0.5

Now, in order to find out exactly what p and r is, we'll need to go through experimentation to find out if Angular Amplitude plays a part in the equation. If we mess around with a pendulum, you'll soon discover that the Angular Amplitude does not affect the oscillation at all. Therefore, p = 0.5, r = 0.

Our final expression would be:
[T] = C [L]^0.5 [LT^-2]^-0.5
T = C(l/g)^0.5

If we scale the length by x, the function would react as:
T(scaled) = C(xl/g)^0.5
T(scaled) = (x)^0.5C(xl/g)^0.5
T(scaled) = (x)^0.5T

This is the furthest we can go without experimentation.