Lect 1 - Powers of 10, Units, Dimensions, Uncertainties, Scaling Arguments

Table of Contents:

A) Introduction to Physical Quantities and Units in Measurement

A.1) Comprehending the Scale of the Universe

In physics, we explore the very small to the very large.
The very small is a fraction of a proton and the very large is the universe itself.

They span 45 orders of magnitude (a 1 with 45 zeroes).
1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

To express measurements quantitatively, we have to introduce "Units".

Length	Time	Mass
m	sec	kg

The definition of these units have evolved during history.

There are many derived units which we use in our daily life for convenience and some are tailored toward specific fields.

For example:

For measuring Length we have:
- centimeters (cm)
- millimeters (mm)
- Kilometers (Km)
- inches (in)
- feet (ft)
- miles (mi)
- and even Astronomical Units (au), which are defined as the mean distance between the Earth and the Sun.
- Light years (ly), which are defined as the distance that light travels in a year.
For measuring Time we have:
- milliseconds (ms)
- microseconds (us)
- days
- weeks
- hours
- centuries
- months
For measuring the Mass we have:
- miligrams (mg)
- pounds (lbm)
- metric tons (t)
- etc.

There are lots of derived units!

Now watch the short video:
"The Powers of Ten" by Kees Boeke

It covers 40 orders of magnitude!

A.2) Fundamental and Derived Quantities

Length (L), Time (T) and Mass (M) are the three fundamental quantities in Physics.

Many other quantifies in physics can be derived from these fundamental quantities.

For Example:
The dimensions of speed is the dimensions of length divided by the dimensions of time

Volume would have the dimensions of length to the power of three.

And we have more examples like density,

or acceleration.

Now that we have defined our units, we can start making measurements.

Important

We shall never forget the uncertainty in our measurements.
Any measurement that you make without the knowledge of its uncertainty is completely meaningless!

#LearnMore about the SI Base quantities

The fundamental quantities in physics, also known as base quantities, are the set of quantities in terms of which all other quantities, known as derived quantities, can be expressed. The International System of Units (SI) identifies seven fundamental physical quantities:

Length - Meter (m)
Mass - Kilogram (kg)
Time - Second (s)
Electric Current - Ampere (A)
Thermodynamic Temperature - Kelvin (K)
Amount of Substance - Mole (mol)
Luminous Intensity - Candela (cd)

These quantities are considered independent of each other, and their units are the basis for other units used in physics for derived quantities like velocity, acceleration, force, etc.

A.3) Demonstration of Measurement Uncertainty (Human Height Variation in Different Positions)

Demo: Imagine that you have noticed that someone who is lying in bed is longer than someone who stands up, and you want to test if this idea is true.

So you build a setup where you can measure a person standing up and a person lying down.

Using a ruler will introduce an uncertainty of 1 mm in the measurements.

And we can show this uncertainty by measuring an aluminum bar (assuming that the length of the bar is the same in both position) in our setup.

These are the results for the Aluminum bar:

If the difference in lengths between lying down and standing up were one foot, we would all know it... But we know that is not the case.

If the difference were only 1 mm, we would never know, because that would be very hard for a person to see and no one would notice.

Therefore, if we want to prove that our idea is true then the difference must be a few centimeters.

And so we can argue that if we can measure the length of a person to 1 mm accuracy that should settle the issue.

These are the results for a person:

The accuracy of 1 mm was more than sufficient to make the case.

If the accuracy of the measurements would have been much less, this measurement would not have been convincing at all.

Remember

Any measurement that you make without the knowledge of its uncertainty is completely meaningless!

B) Scaling Laws in Nature (Galileo’s Insight into the Size of Mammals)

Galieo Galilei asked himself the question: Why are mammals as large as they are and not much larger?

He had a very clever reasoning, he argued that if the mammal becomes to massive, it's bones will break and that was the limiting factor.

B.1) Formulating a Scaling Argument for Animal Femur Sizes

Now let's make a scaling argument:

We could argue that the length of the femur most be proportional to the size of the animal.

We can also argue that the mass of the animal is proportional to the third power of the size (its volume).

Now comes the argument:

Pressure in the femur is proportional to the weight of the animal divided by the cross-section area of the bone.

By removing some constants we get,

If the pressure is higher than a certain level, the bones will break. So in order to stop the pressure from growing to much, we can argue the following:

Now we compare ① and ②

Therefore,

So, what is this result telling us?

If you compare a mouse with an elephant, the elephant is about 100 times larger in size.

So the lenght of the femur in the elephant is 100 times larger than that of the mouse.

But the thickness of the femur would have to be $100^{3 / 2} = 1000$ times larger!

So if the bones grow faster than the size of the animal, there is going to be a point where the thickness of the bones is the same as the height of the bones. And that is biologically not feasible!

So there is a limit somewhere set by this scaling law. And that may have convinced Galileo Galilei on why the largest animals are as large as they are.

Do you know the difference between a Scaling Argument and a Scaling Law?

Scaling Argument: This is a theoretical framework or line of reasoning used to predict how a physical quantity changes as the size of the system changes. It is essentially a way to deduce relationships between quantities based on their dimensional properties and how they should change with scale. For example, in biomechanics, a scaling argument might be used to infer how the strength of a bone changes with the size of an animal.

Scaling Law: This refers to a specific mathematical relationship that describes how certain physical quantities scale with the size of a system. Scaling laws are often derived from scaling arguments and can be empirically tested. They are precise formulations that can be used to predict the behavior of systems when they are scaled up or down. An example is the square-cube law, which states that as an object scales up, its volume grows faster than its surface area.

While a scaling argument involves the process of reasoning to predict how scaling affects various properties, a scaling law is the precise relationship that results from that argument. They are not the same, but a scaling law can be the formal outcome of a scaling argument.

B.2) Testing the Scaling Law for Animal Femur Sizes

Let's bring this idea to a test!

For this test, we will compare the femurs of various mammals:

Raccoon,
horse,
antelope,
opossum,
mouse
and elephant.

First, we will test if the length of the femur is proportional to the size of the different animals, given by:

We can compare the femur of a raccoon and a horse.

A horse is about 4 times bigger than the raccoon. So the length of the femur of a horse must be about 4 times the length of the raccoon. And that is TRUE!

Now let's compare the thickness!

We will test our result:

But instead of comparing the thickness alone, we will be comparing the thickness relative to the length.

With our result, this ratio should be proportional to the square root of the length.

It's time to plot our results, in the horizontal axis we show $l$ in a logarithmic scale and on the vertical axis we show $d_{A V G} / l$
"If Galileo's argument is true, we should see a plot like this:

And here are our results:

Bad news for Galileo... 😭

There is no evidence whatsoever that $d / l$ is really larger for the elephant than for the mouse, the vertical bars indicate the uncertainty in the measurements of thickness, and the uncertainty in length is in the thickness of the red pen (in the horizontal scale).

And here are all the measurements:

Look again at the mouse and look at the elephant.

The elephant is indeed 100 times longer than the mouse. So the first scaling argument ( $S \propto l$ ) hold true.

But when you compare $d / l$ , you see it's all over the place, the ratio for the mouse is really not all that different for the elephant (it was not not $\sqrt{100} = 10$ times larger).

C) Introduction to Dimensional Analysis: A Self-Correcting Methodology

Question: If I drop an apple from a certain height, and I change that height.
What will happen with the time for the apple to fall?

We do another scaling argument and we say that:

If the height is larger, it takes longer for the apple to fall.
If something is more massive, it will probably take less time to fall.
If gravity is stronger, it will take less time to fall

But, how do we know that all of the arguments we made are correct?

Well, one of the benefits of "Dimensional Analysis" is that it is "self-correcting", that is, the procedure, by itself, eliminates the units that are not necessary (but it cannot bring missing units).

Let's start this "Dimensional Analysis" for our example:

The most basic rule of dimensional analysis is that of dimensional homogenity:

Only commensurable quantities (physical quantities having the same dimension) may be compared, equated, added or subtracted.
One may take ratios of in commensurable quantities (quantities with different dimensions), and multiply or divide them.

This means that, the dimensions on the left and right hand side of the equation most be the same.

Now we can determine $α, β$ or\ $γ$ for the equation to be true:

With this result we can only compare two different hights. We can't compute the time for 1 single case.

C.1) Demonstration of the Mass-Independence of Falling Objects (The Falling Apple Experiment)

Demo: We built a setup for dropping apples.

$h_{1} = 3.000 \pm 0.003 m$
$h_{2} = 1.500 \pm 0.003 m$

and we make a prediction of $t_{1} / t_{2}$ .

First we write the ratio $h_{1} / h_{2}$ carrying the uncertainty.

Remember how to carry uncertainty (at A-level)

For addition and subtraction, add the absolute uncertainties directly
For multiplication and division, add the relative uncertainties
For powers and roots, multiply the relative uncertainty by the power.

Now, we make observations:

Time can be measured to an accuracy of 2ms.

This are our results:

Conclusions on our Results: We have demonstrated that the time that it takes for an object to fall is independent of its mass!

Question: Is this dimensional analysis perhaps one that could have been done differently?
Ans: Yes!

C.2) Alternative Approaches to Dimensional Analysis and its Limitations

We could have made different arguments from the very beginning, for example:

If the height is larger, it takes longer for the apple to fall. (same as before)
If something is more massive, it will probably take more time to fall (it doesn't matter because we just showed that time is independent of the mass)
If the mass of the Earth increases, the apple will fall faster (we are no longer using the acceleration due to gravity, we are using the mass of the planet).

The Dimensional Analysis will not work now...

There are no units of time in the right hand side, so we cannot do anything.
End of the story...

Is there something wrong with the analysis that we did?
Is the first one perhaps better than the last one?
Ans: Well, we can't say if it is better or worse. But we know they are different.

We came to the conclusion that the time that it takes for the apple to fall is independent of the mass.

Question: Do we believe that?
Ans: Yes, we do.

On the other hand, there are very prestigious physicists who even nowadays do very fancy experiments and they try to demonstrate that the time for an apple to fall does depend on its mass even though it probably is only very small, if it's true but they try to prove that.

So we do believe that it's independent of the mass.
However, this was not a proof because if you do it with the Mass of the Earth, you get stuck in the analysis.

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Z) 🗃️ Glossary

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