Chapter 1: Basic Physics SE-Units
The Big Idea
Units identify what a specific number represents. For example, the number 42 can be used to represent 42 miles, 42 pounds, or 42 elephants! Without the units attached, the number is meaningless. Also, correct unit cancellation can help you find mistakes when you work out problems.
Key Concepts
- Every calculation and answer to a physics problem must include units. Even if a problem explicitly asks for a speed in meters per second (m/s), the answer is 5 m/s, not 5.
- When you’re not sure how to attack a problem, you can often find the appropriate equation by thinking about which equation will provide an answer with the correct units. For instance, if you are looking to predict or calculate a distance, use the equation where all the units cancel out, with only a unit of distance remaining.
- This book uses SI units (La Système International d’Unités).
- When converting speeds from one set of units to another, remember the following rule of thumb: a speed measured in mi/hr is about a little more than double the value measured in m/s (i.e., 10 m/s is equal to about 20 MPH). Remember that the speed itself hasn’t changed, just our representation of the speed in a certain set of units.
- If a unit is named after a person, it is capitalized. So you write “10 Newtons,” or “10 N,” but “10 meters,” or “10 m.”
- Your weight is simply the force of gravity on you, so pounds are just the English version of Newtons
Key Equations
- 1 meter = 3.28 feet
- 1 mile = 1.61 kilometers
- 1 lb. (1 pound) = 4.45 Newtons
- 1 kg is equivalent to 2.21 lb. on Earth where acceleration of gravity is \begin{align*}9.8 \ m/s^2\end{align*}
An Example of unit conversion:
20 m/s = ? mi/hr
20 m/s (1 mi/1600 m) = .0125 mi/s
.0125 mi/s (60 s/min) = .75 mi/min
.75 mi/min (60 min/hr) = 45 mi/hr
Calculator 101 –TI83
- Arrow Keys: Use the arrow keys to scroll around on the line that your typing your calculation and use the DEL key to delete anything. Use the insert key (type \begin{align*}2^{\text{nd}}\end{align*} then DEL) to add something in.
- Calculator only does what you tell it to: The order of operation is as follows and in order: exponents, multiplication and division, adding and subtracting.
- Some useful tips: The orange key in the upper left corner is very handy. If you hit this button first and then another key, the calculator will operate the command in orange above the key. Here are some good ones
- \begin{align*}\boxed{2^{\text{nd}}}\end{align*} then \begin{align*}\boxed{\text{ENTER}}\end{align*} = ENTRY: the calculator will repeat the last line. If you want the one above it, do it twice. Etc.
- \begin{align*}\boxed{2^{\text{nd}}}\end{align*} then \begin{align*}\boxed{(-)}\end{align*} = ANS: the calculator will put in the answer from last calculation into your equation
- Scientific Notation: On the calculator the best way to represent \begin{align*}3 \times 10^8\end{align*} is to type in 3E8, which means 3 times ten to the eighth power. You get the ‘E’ by hitting the \begin{align*}\boxed{2^{\text{nd}}}\end{align*} then \begin{align*}\boxed{,}\end{align*}.
- Common Problems: Student wants to divide 400 by \begin{align*}2 \pi\end{align*}, but types ‘\begin{align*}400/2 \pi\end{align*}’ into calculator. The calculator reads this as ‘400 divided by 2 and multiplied by \begin{align*}\pi\end{align*}’. What you should do is either ‘\begin{align*}400/(2 \pi)\end{align*}’ or ‘\begin{align*}400/2\end{align*}’ enter then ‘\begin{align*}/\pi\end{align*}’ enter.
- Student wants multiply 12 by the result of 365-90, but types in ‘\begin{align*}12*365-9\end{align*}’. The calculator multiplies 12 by 365 then subtracts 90. What you should do is type in ‘\begin{align*}12*(365-90)\end{align*}’.
- Student types in a long series of operations on the calculator, hits enter and realizes the answer is not right. Solution is to do one operation at a time. For example, instead of typing in ‘\begin{align*}2E3*10/\pi + 45*7 - 5\end{align*}’. Type in \begin{align*}2E3\end{align*}, hit enter then hit ‘\begin{align*}x\end{align*}’ key then ‘10, hit enter, etc. Note that Answer will automatically appear if start a line with an operation (like adding, subtracting, multiplying, dividing)’.
- Make sure you are in degree mode: hit the ‘MODE’ key and then use the arrow to highlight ‘Degree’ and hit ENTER.
- Negative Exponents: You must use the white ‘\begin{align*}(-)\end{align*}’ key for negative exponents.
- Rounding: Round your answers to a reasonable number of digits. If the answer from the calculator is 1.369872654389, but the numbers in the problem are like 21 m/s and 12s, you should round off to 1.4. To report the full number would be misleading (i.e. you are telling me you know something to very high accuracy, when in fact you don’t). Do not round your numbers while doing the calculation, otherwise you’ll probably be off a bit due to the dreaded ‘round off error’.
- Does your answer make sense: Remember you are smarter than the calculator. Check your answer to make sure it is reasonable. For example, if you are finding the height of a cliff and your answer is 0.00034 m. That can’t be right because the cliff is definitely higher than 0.34 mm. Another example, the speed of light is \begin{align*}3 \times 10^8 \ m/s\end{align*} (i.e. light travels 300 million meters in one second). Nothing can go faster than the speed of light. If you calculate the speed of a car to be \begin{align*}2.4 \times 10^{10} \ m/s\end{align*}, you know this is wrong and possibly a calculator typ-o.
Key Applications
The late, great physicist Enrico Fermi used to solve problems by making educated guesses. For instance, say you want to guesstimate the number of cans of soda drank by everybody in San Francisco in one year. You’ll come pretty close if you guess that there are about 800,000 people in S.F., and that each person drinks on average about 100 cans per year. So, 80,000,000 cans are consumed every year. Sure, this answer is wrong, but it is likely not off by more than a factor of 10 (i.e., an “order of magnitude”). That is, even if we guess, we’re going to be in the ballpark of the right answer. That is always the first step in working out a physics problem.
Type of measurement | Commonly used symbols | Fundamental units |
---|---|---|
length or position | \begin{align*}d, x, L\end{align*} | meters (m) |
time | \begin{align*}t\end{align*} | seconds (s) |
velocity | \begin{align*}v\end{align*} | meters per second (m/s) |
mass | \begin{align*}m\end{align*} | kilograms (kg) |
force | \begin{align*}F\end{align*} | Newtons (N) |
energy | \begin{align*}E, K, U, Q\end{align*} | Joules (J) |
power | \begin{align*}P\end{align*} | Watts (W) |
electric charge | \begin{align*}q, e\end{align*} | Coulombs (C) |
temperature | \begin{align*}T\end{align*} | Kelvin (K) |
electric current | \begin{align*}I\end{align*} | Amperes (A) |
electric field | \begin{align*}E\end{align*} | Newtons per Coulomb (N/C) |
magnetic field | \begin{align*}B\end{align*} | Tesla (T) |
Pronunciation table for commonly used Greek letters
\begin{align*}&\mu \ \text{``mu''} & & \tau \ \text{``tau''} & & \Phi \ \text{``phi''}^\ast & & \omega \ \text{``omega''} & & \rho \ \text{``rho''}\\ &\theta \ \text{``theta''} & & \pi \ \text{``pi''} & & \Omega \ \text{``omega''}^\ast & & \lambda \ \text{``lambda''} & & \Sigma \ \text{``sigma''}^\ast \\ &\alpha \ \text{``alpha''} & & \beta \ \text{``beta''} & & \gamma \ \text{``gamma''} & & \Delta \ \text{``delta''}^\ast & & \varepsilon \ \text{``epsilon''}\end{align*}
\begin{align*}^\ast\end{align*}upper case
(a subscript zero, such as that found in \begin{align*}``X_0''\end{align*} is often pronounced “naught” or “not”)
Units and Problem Solving Problem Set
‘Solving a problem is like telling a story.’ -Deb Jensen 2007
For the following problems and all the problems in this book, show the equation you are using before plugging in the numbers, then show each and every step as you work down line by line to get the answer. Equally important, show the units canceling at each step.
Example:
Problem: An \begin{align*}8^{\text{th}}\end{align*} grader is timed to run 24 feet in 12 seconds, what is her speed in meters per second?
Solution:
\begin{align*}D & = rt\\ 24 \ ft & = r(12 \ s)\\ r & = 24 \ ft/12 \ s = 2 \ ft/s\\ r & = 2 \ ft/s \ast (1 m/3.28 \ ft) = 0.61 \ m/s\end{align*}
- Estimate or measure your height.
- Convert your height from feet and inches to meters
- Convert your height from feet and inches to centimeters (100 cm = 1 m)
- Estimate or measure the amount of time that passes between breaths when you are sitting at rest.
- Convert the time from seconds into hours
- Convert the time from seconds into milliseconds (ms)
- Convert the French speed limit of 140 km/hr into mi/hr.
- Estimate or measure your weight.
- Convert your weight in pounds on Earth into a mass in kg
- Convert your mass from kg into \begin{align*}\mu g\end{align*}
- Convert your weight into Newtons
- Find the SI unit for pressure (look it up on internet).
- An English lord says he weighs 12 stones.
- Convert his weight into pounds (you may have to do some research online)
- Convert his weight from stones into a mass in kilograms
- If the speed of your car increases by 10 mi/hr every 2 seconds, how many mi/hr is the speed increasing every second? State your answer with the units mi/hr/s.
- A tortoise travels 15 meters (m) in two days. What is his speed in m/s? mi/hr?
- \begin{align*}80 \ m + 145 \ cm + 7850 \ mm = X \ mm.\end{align*} What is \begin{align*}X\end{align*}?
- A square has sides of length 45 mm. What is the area of the square in \begin{align*}mm^2\end{align*}?
- A square with area \begin{align*}49 \ cm^2\end{align*} is stretched so that each side is now twice as long. What is the area of the square now? Convert to \begin{align*}m^2\end{align*}
- A spacecraft can travel 20 km/s. How many km can this spacecraft travel in 1 hour (h)?
- A dump truck unloads 30 kilograms (kg) of garbage in 40 s. How many kg/s are being unloaded?
- Looking at the units for Electric field (E) in the table at the beginning of the chapter, what do you think the formula is for Force in an electric field?
- Estimate the number of visitors to Golden Gate Park in San Francisco in one year. Do your best to get an answer that is right within a factor of 10. Show your work and assumptions below.
- Estimate the number of plastic bottles of water that are used in one year at Menlo.
Answers:
- We will solve this problem for a person of height 5 ft. 11 in (equivalent to 71/12 fit).
- \begin{align*}71/12 \ ft * (1 \ m/3.28 \ ft) = 1.80 \ m\end{align*}
- The same person is 180 cm
- We will solve this problem assuming 3 seconds between breaths.
- \begin{align*}3 \ seconds = \frac{1}{1200} \ hours\end{align*}
- \begin{align*}3 \times 10^3 \ ms\end{align*}
- 87.5 mi/hr
- (c) If the person weighs 150 lb. this is equivalent to 668 N
- Pascals (Pa), which equals \begin{align*}N/m^2\end{align*}
- 1 stone is equal to 14 pounds.
- 168 lb.
- 76.2 kg
- 5 mi/hr/s
- \begin{align*}0.0000868 \ m/s = 8.7 \times 10^{-5} \ m/s, 1.9 \times 10^{-4} \ mph\end{align*}
- 89,300 mm
- \begin{align*}2025 \ mm^2\end{align*}
- \begin{align*}196 \ cm^2, 0.0196 \ m^2\end{align*}
- 72,000 km/h
- 0.75 kg/s
- \begin{align*}F = qE\end{align*}
- Discuss in class
- Discuss in class