Objectives
- Reinforce your ability to use unit conversions in real-world and scientific contexts.
Section 5.2 Explore
Worksheet 5.2.1 Explore: Unit Conversions
This worksheet is designed to help you explore the concept of unit conversions in physics.
1.
The International Space Station (ISS) uses a water recovery system to recycle wastewater into clean water used for drinking, food preparation, and hygiene. The recovery system can clean \(98\%\) of wastewater into clean water. Suppose each astronaut needs about \(1.00\) gallons of clean water per day. Convert this amount of clean water into meters cubed.
1
en.wikipedia.org/wiki/ISS_ECLSSUseful conversion factors:
\(1\, \text{gallon} = 3.78541\, \text{liters}\)
\(1\, \text{meter}^3 = 1000\, \text{liters}\)
Hint.
Start by converting gallons to liters, then convert liters to cubic meters. For example: \(1.00\, \text{gallons} \left( \frac{3.78541\, \text{liters}}{1\, \text{gallons}} \right)\)
Answer.
\(1\, \text{gallon} = 0.00379\, \text{m}^3\)
Solution.
\(1.00\, \text{gallons} \left( \frac{3.78541\, \text{liters}}{1\, \text{gallons}} \right) \left( \frac{1\, \text{m}^3}{1000\, \text{liters}} \right) = 0.00379\, \text{m}^3\) when rounded to three significant figures.
2.
Astronauts traveling to the International Space Station (ISS) are allowed to take with them personal items with a totatl weight up to \(3.3\, \text{lb}\text{.}\) The items are contained in a personal preference kit (PPK). Convert \(3.3\, \text{lb}\) into grams.
2
en.wikipedia.org/wiki/Personal_preference_kitUseful conversion factors:
\(1\, \text{kg} = 2.20462\, \text{lb}\)
Hint.
Start by converting pounds to kilograms, then convert grams. For example: \(3.3\, \text{lb} \left( \frac{1\, \text{kg}}{2.20462\, \text{lb}} \right)\text{.}\)
Answer.
\(3.3\, \text{lb} = 15\underline{0}0\, \text{g}\)
Solution.
\(3.3\, \text{lb} \left( \frac{1\, \text{kg}}{2.20462\, \text{lb}} \right) \left( \frac{10^3\, \text{g}}{1\, \text{kg}} \right) = 15\underline{0}0\, \text{g}\) when rounded to three significant figures.
3.
The International Space Station (ISS) maintains an internal pressure close to the atmospheric pressure at sea-level on Earth. The pressure in the ISS is held at about \(14.7\, \text{psi}\text{,}\) or pounds per square inch. Convert this pressure to \(\text{Pa}\text{,}\) or pascals.
Useful conversion factors:
\(1\, \text{psi} = 6894.76\, \text{Pa}\)
Hint.
Start by converting psi to Pa using the following conversion factor: \(\left( \frac{6894.76\, \text{Pa}}{1\, \text{psi}} \right)\text{.}\)
Answer.
\(14.7\, \text{psi} = 101000\, \text{Pa}\)
Solution.
\(14.7\, \text{psi} \left( \frac{6894.76\, \text{Pa}}{1\, \text{psi}} \right) = 101000\, \text{Pa}\) when rounded to three significant figures.
4.
The total pressurized volume of the ISS is reported to be \(35491\, \text{ft}^3\text{.}\) Convert this volume into cubic meters \(\text{m}^3\text{.}\)
Useful conversion factors: \(1\, \text{m} = 3.28084\, \text{ft}\)
Hint.
Start by converting feet to meters, then apply the conversion as many times as needed for cubmic meters. For example: \(\left( \frac{1\, \text{m}}{3.28084\, \text{ft}} \right)\text{.}\)
Answer.
\(35491\, \text{ft}^3 = 1005.0\, \text{m}^3\)
Solution.
\(35491\, \text{ft}^3 \left( \frac{1\, \text{m}}{3.28084\, \text{ft}} \right)^3 = 1005.0\, \text{m}^3\) when rounded to five significant figures.
Note that the conversion factor is cubed because we are converting cubic feet to cubic meters. Instead of the shorthand notation by cubing the conversion factor, you could also write it out as: \(35491\, \text{ft}^3 \left( \frac{1\, \text{m}}{3.28084\, \text{ft}} \right) \left( \frac{1\, \text{m}}{3.28084\, \text{ft}} \right) \left( \frac{1\, \text{m}}{3.28084\, \text{ft}} \right) = 1005.0\, \text{m}^3\text{.}\)
Fermi Problems 5.1.2.4 below:
5.
Each time you take a normal breath, your lungs expand by a certain volume. This expansion is roughly equal to the volume of air you inhale during that breath. Estimate how many such breaths would be required for the total lung-expansion volume to equal the entire pressurized volume of the International Space Station (ISS). Consider only the expansion during inhalation. Use question Worksheet Exercise 5.2.1.4 for the ISS volume. All other quantities should be estimated without looking them up.
Hint.
To solve this Fermi problem, start by estimating the volume of air involved in a single normal breath. To estimate this, think about how much air you inhale when you take a relaxed breath and compare it to the size of common containers, such as a water bottle or a soda can.
Then, use the ISS volume, in SI base units \(\text{m}^3\) from question Worksheet Exercise 5.2.1.4 to find how many such breaths would add up to that volume.
It’s ok to sometimes have to look up basic conversion factors if you don’t practice these Fermi estimates often. The more you practice, the more useful conversions you will remember. For this question, it’s ok to look up the conversion between liters and cubic meters if you need to. The conversion is: \(1000\, \text{liters} = 1\, \text{m}^3\text{.}\)
Answer.
It would take about \(10^{6}\) normal breaths for the total lung-expansion volume to equal the pressurized volume of the ISS.
But don’t stop there! Think about how long it would take to reach that many breaths; see the solution for more details on this.
Solution.
Initial thought 1: From question Worksheet Exercise 5.2.1.4, we already have an estimate for the ISS’s total pressurized volume in SI base units. Let’s suppose that earlier we found the ISS volume to be about \(1000\,\text{m}^3\text{.}\) If your estimate was a bit different, that is okay; Fermi problems are all about reasonable approximations. The goal here is to round to something that would be easy to do calculations with without a calculator.
Initial thought 2: To compare this to breaths, we need an estimate for the volume of air involved in a single normal breath; that is, how much your lungs expand when you inhale.
-
Volume of a single normal breath when inhaling.
- Many people have a sense that a normal, relaxed breath is much smaller than a giant “fill your lungs” breath. Here it’s reasonable to assume that a normal breath is probably larger than a can of soda but smaller than a liter sied bottle of water. So a reasonable estimate for a normal breath is somewhere between those to which would be around \(0.5\) liters of air. But, to keep the numbers simple, we will round this to about \(1\) liter of air per breath.
- We also need this in cubic meters to match the ISS volume. So we need to convert \(1\) liter into \(\text{m}^3\text{.}\) Since \(1000\) liters is equal to \(1\,\text{m}^3\text{,}\) that means: \(1\, \text{liter} \left( \frac{1\, \text{m}^3}{10^3\, \text{liter}} \right) = 10^{-3}\, \text{m}^3\text{.}\) Notice the use of scientific notation to help with algebraic manipulation.
- So we can construct a conversion factor that approximates how breaths are related to volume of lung expansion: \(1\, \text{breath} = 10^{-3}\, \text{m}^3\text{.}\)
-
Number of breaths to “match” the ISS volume.
- Our goal is to find how many normal breaths, would add up to the full ISS volume. We can do this with our newly constructed conversion factor: \(1\, \text{breath} = 10^{-3}\, \text{m}^3\text{,}\) and the volume of the ISS, which we are approximating as \(1000\, \text{m}^3\text{,}\) or \(10^3\, \text{m}^3\text{.}\) \(10^3\, \text{m}^3 \left( \frac{1\, \text{breath}}{10^{-3}\, \text{m^3}} \right) = 10^6\, \text{breath}\text{.}\)
So our estimate is that it would take on the order of \(10^{6}\) normal breaths, or about one million breaths for the total lung-expansion volume to equal the pressurized interior volume of the ISS.
If you had chosen \(0.5\) liters per breath instead of \(1\) liter, you would have got about \(2 \times 10^{6}\) breaths. That is still the same order of magnitude, which is exactly what we care about in Fermi problems.
Just for fun, let’s think about how much time it would take for this to happen, afterall that is the spirit of Fermi problems. You can count how many breaths you take in about 30 seconds, then use that to estimate how many breaths you take in a minute, an hour, a day, and so on. If you take about \(12\) to \(18\) breaths per minute, we treat that is \(10^1\, \frac{\text{breath}}{min}\text{.}\) So to figure how how many breaths per day you take, we can use the conversions \(60\, \text{min} = 1\, \text{hr}\text{,}\) but in the spirit of Fermi problems, we can round that to \(10^2\, \text{min} = 1\, \text{hr}\text{,}\) likewise \(24\, \text{hr} = 1\, \text{day}\text{,}\) which we can round to \(10^1\, \text{hr} = 1\, \text{day}\text{.}\) So we have: \(10^6\, \text{breath} \left( \frac{1\, \text{min}}{10^1\, \text{breath}} \right) \left( \frac{1\, \text{hr}}{10^2\, \text{min}} \right) \left( \frac{1\, \text{day}}{10^1\, \text{hr}} \right) = 10^2\, \text{day}\text{.}\) So it would take about 100 days of continuous, normal breathing for the total expansion of your lungs to equal the entire pressurized volume of the ISS.
