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1.1: Units and Problem Solving

Difficulty Level: At Grade Created by: CK-12

Units identify what a specific number refers to. For instance, the number 42 can be used to represent 42 miles, 42 pounds, or 42 elephants! Numbers are mathematical objects, but units give them physical meaning. Keeping track of units can help you avoid mistakes when you work out problems.

Key Concepts

  • Every 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 approach a problem, you can often get insight by considering how to obtain the units of the desired result by combining the units of the given variables. For instance, if you are given a distance (in meters) and a time (in hours), the only way to obtain units of speed (meters/hour) is to divide the distance by the time. This is a simple example of a method called dimensional analysis, which can be used to find equations that govern various physical situations without any knowledge of the phenomena themselves.
  • This textbook uses SI units (La Système International d’Unités), the most modern form of the metric system.
  • When converting speeds from metric to American units, remember the following rule of thumb: a speed measured in mi/hr is about 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.”
  • Vectors are arrows that represent quantities with direction. In this textbook, vectors will be written in bold. For instance, the force vector will be written as \begin{align*}F\end{align*}F in this textbook. Your teacher will likely use\begin{align*} \textstyle{\vec{\textstyle{F}}} \end{align*}F to represent vectors. Don’t let this confuse you: \begin{align*} \vec{F} \end{align*}F represents the same concept as \begin{align*}F\end{align*}F.
  • Vectors can be added together in a simple way. Two vectors can be moved (without changing their directions) to become two legs of a parallelogram. The sum of two vectors is simply the diagonal of the parallelogram:

Key Equations

\begin{align*} 1\ \text{meter} & = 3.28\ \text{feet} && \\ 1\ \text{mile} & = 1.61 \text{~kilometers} && \\ 1\ \text{lb. (1\ pound)} & = 4.45\ \text{Newtons}\end{align*}

1 meter1 mile1 lb. (1\ pound)=3.28 feet=1.61~kilometers=4.45 Newtons

Key Applications

The late great physicist Enrico Fermi used to solve problems by making educated guesses. Say you want to guesstimate the number of cans of soda drunk 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 one person drinks on average about 100 cans per year. So, around 80,000,000 cans are consumed every year. Sure, this answer is not exactly right, but it is likely not off by more than a factor of 10 (i.e., an “order of magnitude”). That is, even though we guessed, we’re going to be in the ballpark of the right answer. This is often 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*}d,x,L meters \begin{align*}(m)\end{align*}(m)
time \begin{align*} t \end{align*}t seconds \begin{align*}(s)\end{align*}(s)
velocity or speed \begin{align*} v, u \end{align*}v,u meters per second \begin{align*}(m/s)\end{align*}(m/s)
mass \begin{align*} m \end{align*}m kilograms \begin{align*}{(kg)}\end{align*}(kg)
force \begin{align*}F\end{align*}F Newtons \begin{align*}{(N)}\end{align*}(N)
energy \begin{align*} E, K, U, Q \end{align*}E,K,U,Q Joules \begin{align*}(J)\end{align*}(J)
power \begin{align*} P \end{align*}P Watts \begin{align*}(W)\end{align*}(W)
electric charge \begin{align*} q, e \end{align*}q,e Coulombs \begin{align*}(C)\end{align*}(C)
temperature \begin{align*} T \end{align*}T Kelvin \begin{align*}(K)\end{align*}(K)
electric current \begin{align*} I \end{align*}I Amperes \begin{align*}(A)\end{align*}(A)
electric field \begin{align*}E\end{align*}E Newtons per Coulomb \begin{align*}(N/C)\end{align*}(N/C)
magnetic field \begin{align*}B\end{align*}B Tesla \begin{align*}(T)\end{align*}(T)
magnetic flux \begin{align*}\Phi\end{align*}Φ Webers \begin{align*}{(Wb)}\end{align*}(Wb)

Pronunciation table for commonly used Greek letters

\begin{align*}&\mu~~\text{mu} & & \tau~~\text{tau} & & \Phi~~\text{Phi}^* & & \omega~~\text{omega} & & \rho~~\text{rho}\\ &\theta~~\text{theta} & & \pi~~\text{pi} & & \Omega~~\text{Omega}^* & & \lambda~~\text{lambda} & & \Sigma~~\text{Sigma}^*\\ &\alpha~~\text{alpha} & & \beta~~\text{beta} & & \gamma~~\text{gamma} & & \Delta~~\text{Delta}^* & & \epsilon~~\text{epsilon}\end{align*}

μ  muθ  thetaα  alphaτ  tauπ  piβ  betaΦ  PhiΩ  Omegaγ  gammaω  omegaλ  lambdaΔ  Deltaρ  rhoΣ  Sigmaϵ  epsilon

\begin{align*}^*\text{upper case (a subscript zero, such as that found in}\ x_0 \ \text{is often pronounced ``naught'' or ``not'')}\end{align*}upper case (a subscript zero, such as that found in x0 is often pronounced ``naught'' or ``not'')

Units and Problem Solving Problem Set

  1. Estimate or measure your height.
    1. Convert your height from feet and inches to meters.
    2. Convert your height from feet and inches to centimeters \begin{align*}(100 \;\mathrm{cm} = 1 \;\mathrm{m})\end{align*}(100cm=1m)
  2. Estimate or measure the amount of time that passes between breaths when you are sitting at rest.
    1. Convert the time from seconds into hours
    2. Convert the time from seconds into milliseconds = (ms)
  3. Convert the French speed limit of 140 km/hr into mi/hr.
  4. Estimate or measure your weight.
    1. Convert your weight in pounds into a mass in kg
    2. Convert your mass from kg into \begin{align*}\mu g\end{align*}μg
    3. Convert your weight into Newtons
  5. Find the \begin{align*}SI\end{align*}SI unit for pressure.
  6. An English lord says he weighs 12 stone.
    1. Convert his weight into pounds (you may have to do some research online)
    2. Convert his weight in stones into a mass in kilograms
  7. 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.
  8. A tortoise travels 15 meters (m) west, then another 13 centimeters (cm) west. How many meters total has she walked?
  9. A tortoise, Bernard, starting at point A travels 12 m west and then 150 millimeters (mm) east. How far west of point \begin{align*}A\end{align*}A is Bernard after completing these two motions?
  10. \begin{align*}80 \;\mathrm{m} + 145 \;\mathrm{cm} + 7850 \;\mathrm{mm} = X\ \;\mathrm{mm} \end{align*}80m+145cm+7850mm=X mm. What is\begin{align*} X \end{align*} ?
  11. A square has sides of length 45 mm. What is the area of the square in \begin{align*}\;\mathrm{mm}^2\end{align*}?
  12. A square with area \begin{align*}49 \;\mathrm{cm}^2\end{align*} is stretched so that each side is now twice as long. What is the area of the square now? Include a sketch.
  13. A rectangular solid has a square face with sides 5 cm in length, and a length of 10 cm. What is the volume of the solid in \begin{align*}\;\mathrm{cm}^3\end{align*}? Sketch the object, including the dimensions in your sketch.
  14. As you know, a cube with each side 4 m in length has a volume of 64 \begin{align*}\;\mathrm{m}^3\end{align*}. Each side of the cube is now doubled in length. What is the ratio of the new volume to the old volume? Why is this ratio not simply 2? Include a sketch with dimensions.
  15. What is the ratio of the mass of the Earth to the mass of a single proton? (See equation sheet.)
  16. A spacecraft can travel 20 km/s. How many km can this spacecraft travel in 1 hour (h)?
  17. A dump truck unloads 30 kilograms (kg) of garbage in 40 s. How many kg/s are being unloaded?
  18. The lengths of the sides of a cube are doubling each second. At what rate is the volume increasing?
  19. Estimate the number of visitors to Golden Gate Park in San Francisco in one year. Do your best to get an answer that is correct within a factor of 10.
  20. Estimate the number of water drops that fall on San Francisco during a typical rainstorm.
  21. What does the formula \begin{align*} a = \frac{F} {m} \end{align*} tell you about the units of the quantity \begin{align*} a \end{align*} (whatever it is)?
  22. Add the following vectors using the parallelogram method.

Answers to Selected Problems

    1. A person of height 5 ft. 11 in. is 1.80 m tall
    2. The same person is 180 cm
    1. \begin{align*}3 \;\mathrm{seconds} = 1/1200 \;\mathrm{hours}\end{align*}
    2. \begin{align*}3 \times 10^3 \;\mathrm{ms}\end{align*}
  1. 87.5 mi/hr
  2. c. if the person weighs 150 lb. this is equivalent to 668 N
  3. Pascals (Pa), which equals \begin{align*}\;\mathrm{N/m}^2\end{align*}
  4. 168 lb., 76.2 kg
  5. 5 mi/hr/s
  6. 15.13 m
  7. 11.85 m
  8. 89,300 mm
  9. f. \begin{align*}2025 \;\mathrm{mm}^2\end{align*}
  10. b. \begin{align*}196 \;\mathrm{cm}^2\end{align*}
  11. c. \begin{align*} 250 \;\mathrm{cm}^3\end{align*}
  12. 8:1, each side goes up by 2 cm, so it will change by \begin{align*}2^3\end{align*}
  13. \begin{align*}3.5 \times 10^{51}:1\end{align*}
  14. 72,000 km/h
  15. 0.75 kg/s
  16. \begin{align*} 8 \times 2^N \;\mathrm{cm}^3/\;\mathrm{sec}\end{align*}; \begin{align*}N\end{align*} is for each second starting with \begin{align*}0\end{align*} seconds for \begin{align*}8 \;\mathrm{cm}^3\end{align*}
  17. About 12 million
  18. About \begin{align*}1 \frac{1}{2}\end{align*} trillion \begin{align*}(1.5 \times 10^{12})\end{align*}
  19. \begin{align*}[\mathrm{a}] = \;\mathrm{N/kg} = \;\mathrm{m/s}^2\end{align*}
  20. .

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