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Learning Introductory Physics with Activities

Paul J. Emigh, Rebecka Tumblin, Kathryn Hadley, Danielle Skinner

Section 1.8 Sensemaking: Units and Numbers

The measurement of any quantity is reported as a number in terms of some standard, physical unit. For example, you might measure the length of an object in a physics lab to be \(17.4\text{.}\) Without a specific unit of measurement, the number itself is almost meaningless. You might mean the length is \(17.4 \mathrm{~centimeters}\text{,}\) \(17.4 \mathrm{~inches}\text{,}\) or \(17.4 \mathrm{~miles}\text{.}\)
In science, we use the SI unit system (le Systéme Internationale d’Unités), commonly referred to as the metric system 1 . Within the metric system, all quantities will be given in terms of base units, such as the three below, which are typically written using non-italic letters 2 .
Units for physical quantities can be divided into two categories: base units and derived units. Examples of derived units, which are often given their own name, include force (the newton) and energy (the joule), each of which is a specific combination of the base units of length, time and mass.
Time: The SI unit of time is the second. The abbreviation for the second is lower case s.
Length: The SI unit of length is the meter. The abbreviation for the meter is lower case m.
Mass: The SI unit of mass is the kilogram. The abbreviation for the kilogram is lower case kg.
Time: Throughout most of history, the standard of time was based on the mean solar day. However, scientists needed a way to make incredibly high precision measurements of time and so physicists began developing a device called an atomic clock. Atomic clocks work on the principle that subatomic particles called electrons will move back and forth between different energy states when exposed to certain frequencies of radiation. The United States’ standard atomic clock is said to be so accurate that in 30 million years it would neither gain nor lose a second. One second is defined as the time it takes for 9,192,631,770 oscillations of a radio wave absorbed by the cesium-133 atom.
Length: One meter is the distance light can travel in a vacuum during 1/299,792,458 of a second.
Mass: For many years, one kilogram was the mass of a polished platinum-iridium cylinder, the international standard kilogram, which is stored in Paris, France. Since 2019, the kilogram has been defined even more precisely using the above definitions for length and time along with a fundamental quantum mechanical property of the universe called Planck’s constant.
A small cylindrical object inside a glass container.
Figure 1.8.3. An image of the kilogram standard held in Paris, France.
Checking units or numbers are two of the easiest and most powerful ways to make sense of something in physics. When you make sense of something, your objective is evaluate whether or not what physics says about a context aligns with your physical understanding and reasoning about that context, following the Sensemaking Steps.
Suppose that we have derived the escape speed of a rocket from a planet of mass \(M\) and radius \(R\) (escape speed is the speed at which an object must reach to escape the gravitational pull): \(v_{esc} = \sqrt{\frac{2GM}{R}}\text{.}\) As a speed, \(v_{esc}\) should have units like m/s (distance divided by time).
Here, \(G\) represents the universal gravitational constant with units \(\frac{\text{m}^3}{\text{s}^2\text{kg}}\text{.}\) Since there is an equal sign, this implies that the base units of either side of the equal sign must be the same. If they are not, then we have made a mistake in our derivation of the escape speed. We can perform a unit analysis to determine if we have the same base units for each side of the equal sign.
\(\frac{\text{m}}{\text{s}} \stackrel{?}{=} \sqrt{\frac{\frac{\text{m}^3}{\text{s}^2\text{kg}}\text{kg}}{\text{m}}}=\sqrt{\frac{\text{m}^2}{\text{s}^2}}=\frac{\text{m}}{\text{s}} \checkmark\)
We can see that the physical quantities under the square root indeed have base units of [L] divided by a time [T] and so our equation holds true. Observe that we followed the sensemaking steps outlined above: we chose a strategy (checking units), determined what units we should get, found the units we actually have, and compared the two to see that they are the same!
Physics uses the word dimensions to refer to the class of unit for a quantity. For example, speed can be measured in units of km/hr, m/s, or mi/hr but the base dimensions, represented in square brackets, are always a length [L] divided by a time [T].
Consider the example from the start of this section, where you measured the length of an object in a physics lab to be \(17.4 \mathrm{~cm}\text{.}\) You may want to compare this number to the length of other objects that you might expect to find in a physics lab. For example, you might expect that many objects in a physics lab can be held in one hand; you would then want to compare the \(17.4 \mathrm{~cm}\) measurement to the size of a human hand to evaluate its reasonableness. The search term "human hand size" results in a measurement of between \(6.8\) and \(7.6 \mathrm{~in}\text{,}\) which you might approximate to about \(7 \mathrm{~in}\) (note: when checking numbers, you do not need to be precise, just approximate). Converting this \(7 \mathrm{~in}\) to SI units gives approximately \(17.8 \mathrm{~cm}\text{.}\) This is very close to the object you measured in the lab, so it seems like this measurement was pretty reasonable! It is also worth noting here that length is usually given as a positive number, which this measurement is.
You might be asking yourself: what would an unreasonable answer have looked like? Perhaps your measurement had been \(174 \mathrm{~cm}\) instead of \(17.4 \mathrm{~cm}\text{.}\) This is only about ten times bigger, and still seems like a reasonable measurement for a physics lab (something about the height of a person). On the other hand, measuring something like \(17.4 \mathrm{~km}\) would be fairly unreasonable for an ordinary physics lab: 100,000 times bigger than a person’s hand!

Exercises Units Activities

1. Practice Identifying Units.

Below is a list of symbols we commonly use in physics to represent physical quantities. For each symbol, indicate the units you would expect for the corresponding quantity.
  1. \(t\) (time)
  2. \(d\) (distance)
  3. \(m\) (mass)
  4. \(t_f\) (final time)
  5. \(\vec{r}\) (position)
  6. \(A\) (area)
Answer.
  1. s (seconds)
  2. m (meters)
  3. kg (kilograms)
  4. s (seconds)
  5. m (meters)
  6. m\(\mathrm{^2}\) (square meters)

2. Practice Checking Units.

Your friend has developed a new engine that creates carbon dioxide gas. Your friend shows you an equation they created relating the volume of gas created (\(V\)) to the width of the engine (\(w\)), the amount of time the engine has been running (\(\Delta t\)), and the mass of a single molecule of carbon dioxide (\(M\)): \(V = k\frac{w^2\Delta t}{M}\text{.}\) Use what you know about the other symbols in the problem to determine the units of \(k\text{.}\)
Answer.
\(\frac{\text{mkg}}{\text{s}}\)

3. Practice Identifying an Error with Units.

While looking through a mathematics textbook, you come across the following equation for the surface area of a cylinder: \(A = 2\pi rh^2 + 2\pi r^2\text{.}\) You suspect this equation must have a mistake in it. How can you use units to tell what the mistake is? How could you use units to fix the mistake?
Solution.
When you add two quantities together, they must have the same units! The numbers \(2\) and \(\pi\) are unitless, but \(rh^2\) has units of cubic meters while \(r^2\) only has units of square meters. Additionally, since area \(A\) should have units of square meters, you can tell that the first term is the one that is wrong: you probably need to get rid of either the \(r\) or one of the factors of \(h\text{.}\)

Exercises Numbers Activities

1. Practice Identifying Numbers.

Below is a list of numbers with units. For each number, find a real-world object or system with a measurement that is reasonably close 3 .
  1. \(\displaystyle 900,000 \mathrm{~kg}\)
  2. \(\displaystyle 25 \mathrm{~\mu m}\)
  3. \(\displaystyle 3.2 \times 10^7 \mathrm{~s}\)
  4. \(\displaystyle 110 \mathrm{~m}\)
  5. \(\displaystyle 60 \mathrm{~mg}\)
  6. \(\displaystyle 0.3 \mathrm{~s}\)

2. Practice Checking Numbers and Identifying Errors.

Imagine that you came upon the following numbers while reading an article. Try to identify whether or not each number is reasonable without directly looking up the quantity in question.
  1. It takes light around \(8.5\) minutes to reach Earth from the Sun
  2. A semitruck and its loaded trailer is measured to have a mass of \(500 \mathrm{~kg}\)
  3. The distance from the Earth to the Moon is \(384\times10^6 \mathrm{~\mu m}\text{.}\)
  4. The mass of a pea is \(0.2 \mathrm{~g}\)
  5. An Olympic marathon is \(42 \mathrm{~km}\)
  6. An Orca can hold its breath underwater for \(10,000 \mathrm{~s}\)
In the united States, we have an entire government agency dedicated to upholding standards of measurement
prefixes can be used as an easy way to express units that are substantially different
\(\mu\) is the symbol for the prefix micro-; you can search for how big it is
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