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# Metric Units

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Practice Metric Units
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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 $d, x, L$ meters (m)
time $t$ seconds (s)
velocity or speed $v, u$ meters per second (m/s)
mass $m$ kilograms (kg)
force $\mathbf{F}$ Newtons (N)
energy $E, K, U, Q$ Joules (J)
power $P$ Watts (W)
electric charge $q, e$ Coulombs (C)
temperature $T$ Kelvin (K)
electric current $I$ Amperes (A)
electric field $\mathbf{E}$ Newtons per Coulomb (N/C)
magnetic field $\mathbf{B}$ Tesla (T)

#### Prefixes

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

#### Greek Letters

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

Two very common Greek letters are $\Delta$ and $\Sigma$ . $\Delta$ is used to indicate that we should use the change or difference between the final and initial values of that specific variable. $\Sigma$ 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, $x$ , is doubling every second. Therefore after 1 second it becomes $2{x}$ . The volume of the first cube of side ${x}$ is ${x}\times{x}\times{x}={{x}^3}$ . The volume of the second cube of side $2{x}$ is $2{x}\times2{x}\times2{x}={8{x}^3}$ . The ratio of the second volume to the first volume is ${8{x}^3}/{{x}^3}=8$ . Thus the volume is increasing by a factor of 8 every second.

### Explore More

1. A tortoise travels $15$ meters $\;\mathrm{(m)}$ west, then another $13$ centimeters $\;\mathrm{(cm)}$ west. How many meters total has she walked?
2. A tortoise, Bernard, starting at point A travels $12 \;\mathrm{m}$ west and then $150$ millimeters $\;\mathrm{(mm)}$ east. How far west of point $A$ is Bernard after completing these two motions?
3. $80 \;\mathrm{m} + 145 \;\mathrm{cm} + 7850 \;\mathrm{mm} = X\ \;\mathrm{mm}$ . What is $X$ ?
4. A square has sides of length $45 \;\mathrm{mm}$ . What is the area of the square in $\;\mathrm{mm}^2$ ?
5. A square with area $49 \;\mathrm{cm}^2$ 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 $5 \;\mathrm{cm}$ in length, and a length of $10 \;\mathrm{cm}$ . What is the volume of the solid in $\;\mathrm{cm}^3$ ? Sketch the object, including the dimensions in your sketch.
7. As you know, a cube with each side $4 \;\mathrm{m}$ in length has a volume of $64 \;\mathrm{m}^3$ . 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.
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 $20 \;\mathrm{km/s}$ . How many km can this spacecraft travel in 1 hour $\;\mathrm{(h)}$ ?

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