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Metric Units

A system of measurement. Basic units are the meter, the second, and the kilogram.

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Metric Units

Students will learn about the metric system and how to convert between metric units.

Frequently Used Measurements, Greek Letters, and Prefixes

Measurements

Types of Measurements
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 or speed \begin{align*} v, u \end{align*} meters per second (m/s)
mass \begin{align*} m \end{align*} kilograms (kg)
force \begin{align*}\mathbf{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*}\mathbf{E} \end{align*} Newtons per Coulomb (N/C)
magnetic field \begin{align*}\mathbf{B} \end{align*} Tesla (T)

Prefixes

SI prefix In Words Factor
nano (n) billionth \begin{align*}1*10^{-9}\end{align*}
micro (µ) millionth \begin{align*}1*10^{-6}\end{align*}
milli (m) thousandth \begin{align*}1*10^{-3}\end{align*}
centi (c) hundreth \begin{align*}1*10^{-2}\end{align*}
deci (d) tenth \begin{align*}1*10^{-1}\end{align*}
deca (da) ten \begin{align*}1*10^{1}\end{align*}
hecto (h) hundred \begin{align*}1*10^{2}\end{align*}
kilo (k) thousand \begin{align*}1*10^{3}\end{align*}
mega (M) million \begin{align*}1*10^{6}\end{align*}
giga (G) billion \begin{align*}1*10^{9}\end{align*}

Greek Letters

Frequently used Greek letters.
\begin{align*} \mu \end{align*} “mu” \begin{align*} \tau \end{align*} “tau” \begin{align*} \Phi \end{align*} “Phi”* \begin{align*} \omega \end{align*} “omega” \begin{align*} \rho \end{align*} “rho”
\begin{align*} \theta \end{align*} “theta” \begin{align*} \pi \end{align*} “pi ” \begin{align*} \Omega \end{align*} “Omega”* \begin{align*} \lambda \end{align*} “lambda” \begin{align*} \Sigma \end{align*} “Sigma”*
\begin{align*} \alpha \end{align*} “alpha” \begin{align*} \beta \end{align*} “beta” \begin{align*} \gamma \end{align*} “gamma” \begin{align*} \Delta \end{align*} “Delta”* \begin{align*} \epsilon \end{align*} “epsilon”

Two very common Greek letters are \begin{align*}\Delta\end{align*} and \begin{align*}\Sigma\end{align*} . \begin{align*}\Delta\end{align*} is used to indicate that we should use the change or difference between the final and initial values of that specific variable. \begin{align*}\Sigma\end{align*} denotes the sum or net value of a variable.

Guidance
  • 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.
  • 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.”
  • Metric units use a base numbering system of 10. Thus a centimeter is ten times larger than a millimeter. A decimeter is 10 times larger than a centimeter and a meter is 10 times larger than a decimeter. Thus a meter is 100 times larger than a centimeter and 1000 times larger than a millimeter. Going the other way, one can say that there are 100 cm contained in a meter.

Example 1

Question: Convert 2500 m/s into km/s

Solution: A km (kilometer) is 1000 times bigger than a meter. Thus, one simply divides by 1000 and arrives at 2.5 km/s

Example 2

Question: The lengths of the sides of a cube are doubling each second. At what rate is the volume increasing?

Solution:The cube side length, \begin{align*}x\end{align*}, is doubling every second. Therefore after 1 second it becomes \begin{align*}2{x}\end{align*}. The volume of the first cube of side \begin{align*}{x}\end{align*} is \begin{align*}{x}\times{x}\times{x}={{x}^3}\end{align*}. The volume of the second cube of side \begin{align*}2{x}\end{align*} is \begin{align*}2{x}\times2{x}\times2{x}={8{x}^3}\end{align*}. The ratio of the second volume to the first volume is \begin{align*}{8{x}^3}/{{x}^3}=8\end{align*}. Thus the volume is increasing by a factor of 8 every second.

Watch this Explanation

Time for Practice

  1. A tortoise travels \begin{align*}15\end{align*} meters \begin{align*}\;\mathrm{(m)}\end{align*} west, then another \begin{align*}13\end{align*} centimeters \begin{align*}\;\mathrm{(cm)}\end{align*} west. How many meters total has she walked?
  2. A tortoise, Bernard, starting at point A travels \begin{align*}12 \;\mathrm{m}\end{align*} west and then \begin{align*}150\end{align*} millimeters \begin{align*}\;\mathrm{(mm)}\end{align*} east. How far west of point \begin{align*}A\end{align*} is Bernard after completing these two motions?
  3. \begin{align*}80 \;\mathrm{m} + 145 \;\mathrm{cm} + 7850 \;\mathrm{mm} = X\ \;\mathrm{mm} \end{align*}. What is\begin{align*} X \end{align*} ?
  4. A square has sides of length \begin{align*}45 \;\mathrm{mm}\end{align*}. What is the area of the square in \begin{align*}\;\mathrm{mm}^2\end{align*}?
  5. 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.
  6. A rectangular solid has a square face with sides \begin{align*}5 \;\mathrm{cm}\end{align*} in length, and a length of \begin{align*}10 \;\mathrm{cm}\end{align*}. What is the volume of the solid in \begin{align*}\;\mathrm{cm}^3\end{align*}? Sketch the object, including the dimensions in your sketch.
  7. As you know, a cube with each side \begin{align*}4 \;\mathrm{m}\end{align*} in length has a volume of \begin{align*}64 \;\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 \begin{align*}2\end{align*}? Include a sketch with dimensions.
  8. What is the ratio of the mass of the Earth to the mass of a single proton? (See equation sheet.)
  9. A spacecraft can travel \begin{align*}20 \;\mathrm{km/s}\end{align*}. How many km can this spacecraft travel in 1 hour \begin{align*}\;\mathrm{(h)}\end{align*}?

Answers

  1. \begin{align*}15.13 \;\mathrm{m}\end{align*}
  2. \begin{align*}11.85 \;\mathrm{m}\end{align*}
  3. \begin{align*}89,300 \;\mathrm{mm}\end{align*}
  4. \begin{align*}2025 \;\mathrm{mm}^2\end{align*}
  5. \begin{align*}196 \;\mathrm{cm}^2\end{align*}
  6. \begin{align*} 250 \;\mathrm{cm}^3\end{align*}
  7. \begin{align*}8:1,\end{align*} each side goes up by \begin{align*}2 \;\mathrm{cm}\end{align*}, so it will change by \begin{align*}2^3\end{align*}
  8. \begin{align*}3.5 \times 10^{51}:1\end{align*}
  9. \begin{align*} 72,000 \;\mathrm{km/h}\end{align*}

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