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# Metric Units

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

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Practice Metric Units
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Estimated18 minsto complete
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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 d,x,L\begin{align*} d, x, L \end{align*} meters (m)
time t\begin{align*} t \end{align*} seconds (s)
velocity or speed v,u\begin{align*} v, u \end{align*} meters per second (m/s)
mass m\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.

### Explore More

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*}?

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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