Using Dimensional Analysis for Metric Conversions



Using Dimensional Analysis for Metric Conversions

Step 1-Memorize the following chart (Gee, many kids have died by doing conversions. Metric, murderously numbing.)

|1 G(base) = 1 000 000 000 (base) |

|1 M(base) = 1 000 000 (base) |

|1 k(base) = 1 000 (base) |

|1 h(base) = 100 (base) |

|1 da(base) = 10 (base) |

|1 (base) = 1 (base) |

|10 d(base) = 1 (base) |

|100 c(base) = 1 (base) |

|1000 m (base) = 1 (base) |

|1 000 000 µ(base) = 1(base) |

|1 000 000 000 n(base) = 1(base) |

Step 2 – Learn to use the table above. To find the relationship to be used as a conversion factor, replace base with any metric unit. (second, meter, gram, liter, etc) Then use the table to write down the relationship.

For example to convert Mm to m, the relationship above is

1 M(base) = 1 000 000 (base)

That becomes

1 Mm = 1 000 000 m

1. Practice- Write the relationships for the following conversions

a. cm to m

b. m to µm

c. ns to s

d. kg to g

e. L to mL

Step 3- Turn the relationships into conversion factors. Each of the relationships above can become one of two conversion factors.

1 = 1 Mm = 1 000 000 m

1 000 000 m Mm

2. Practice – Write the two conversion factors for the relationships in a-e above

Step 4- Use the conversion factors to do conversions. A conversion from cm to m requires one step. You can identify 1 step conversions because only one of the units has a prefix. A conversion from cm to µm requires two steps. You can identify a 2 step conversion because both of the units have a prefix.

To convert from cm to m - use the relationship 100 cm = 1 m, make the conversion factors

Desired unit is left uncanceled

45 cm 1 m = 0.45 m

100 cm

• Put the unit you want to cancel on the opposite side of the bar

• Multiply by the numbers on the top of the bar and divide by the numbers below the bar

To convert from cm to µm (two steps because there are two prefixes)

□ Identify the relationships between cm and m, and between µm and m

▪ Find the path you will take to make the conversion (cm to m to µm)

100 cm = 1 m 1000 000 µm = 1 m

□ Turn the relationships into conversion factors

□ Insert the conversion factors to cancel the units

□ Multiply the top numbers and divide by the bottom numbers

Example

45 cm 1 m 1000 000 µm = 450000 µm

100 cm 1 m

3. Practice-

a. convert 1.52 x 105 ns to s (follow the steps above)

b. convert 3.51 x 10-6 kg to mg

Making time conversions

□ Determine the time relationships needed to make the conversions

▪ Find the path you will take to make the conversion

□ Change the relationships into conversion factors

□ Insert the conversion factors to cancel the units

□ Multiply the top numbers and divide by the bottom numbers

□ Determine the correct number of significant figures and round

Example

Convert 1.5 x 106 s into days

Path s to min to hr to days

Relationships

60 s = 1 min 60 min = 1 hr 24 hr = 1 day

1.5 x 108 s 1 min 1 hr 1 day

= 1736.11 days only 2 sig figs allowed

60 s 60 min 24 hr

1700 days final answer

4. Practice

a. convert 5.76 x 108 s to years

b. convert 8.553 x 10-2 years to seconds

Making metric conversions on both the top and the bottom and when the unit is raised to a power

□ Determine the relationships needed to make the top conversion and the bottom conversion

□ Do the conversions in steps

□ Cancel the units raised to a power

Example

Convert 45 m/s2 to km/min2 (notice the m to km is one conversion and s2 to min2 is another conversion)

Relationships 1000 m = 1 km 60 s = 1 min

45 m 1 km 60 s 60 s = 162 km only 2 sig figs allowed

s2 1000 m 1 min 1 min min2

160 km/min2 final answer

place the units properly, convert the top unit, then the bottom unit.

5. Practice

a. convert 4.56 g/cm3 to kg/m3

b. convert 78.4 g/mL to kg/L

Applying dimensional analysis to word problems

□ Read the problem carefully to determine the starting and ending units. Look for the actual question.

□ Determine if the starting and ending units are for the same dimension. ( Is it asking to change from mass to volume, or from money to mass?) If they are not, look for a relationship between those dimensions in the problem.

□ Find a pathway from the starting units to the final units

□ Determine all of the relationships

□ Write the starting value with the units and cancel the units

□ Evaluate how many significant figures are allowed and round answer

Example:

Gold sells for $815/ounce. Considering that there are 16 ounces in a pound and 454 g in a pound, how many milligrams of gold could you buy for 10 cents?

□ The underlined section is the question. Convert 10 cents to mg of gold.

□ The price of the gold gives a relationship between $ and ounces (cents can be converted to $ and ounces can be converted to mg)

□ Pathways cents to $ to ounces to grams to mg

□ Relationships

▪ $1 = 100 cents 16 oz = 1 lb = 454 g 1 g = 1000 mg $815 = 1 oz gold

10 cent $1 1 oz gold 1 lb 454 g 1000 mg = 3.4815 mg rounded to 3 sig fig

100 cents $815 16 oz 1 lb 1 g 3.48

3 sig figs

6. Practice

a. The density of aluminum is 2.70 g/cm3. What is the mass in kg of a block of aluminum that measures 5.00 cm x 8.00 cm x 3.0 cm

b. A large river flows at the rate of 6.5 x 10 5 cm3/s into a rectangular shaped boat lock 25 m long, 45 m wide and 20 m high. How many minutes before the lock would be filled to the top with water? (Start with the volume of the lock)

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