Physics - Sig Figs and Railroad Tracks Continue

OBJECTIVES:

I will be able to successfully identify significant figures for MOST values I observe today.

I will be able to setup basic railroad track conversions during class today

 

WORDS O' THE DAY:

• Sig Figs

WORK O' THE DAY:

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Opening Question Answers:

51.88" = 5.188 x 10¹ inches

• 51.88 x 10⁰ inches also works
• .5188 x 10² inches also works

However in physics the generally preferred method is to use only one digit to the left of the decimal. I prefer that but won't get TOO dogmatic about it

Unit Conversions using railroad tracks method. Remember, I will INSIST that you learn this because I know how amazingly powerful it is and I will I had learned this myself....way back in the day:

Step #1 Write the value to be converted as a fraction. In this case there is no denominator so we leave that blank:

51.88 inches

Step #2 Write the unit type in the denominator of the second track (this is critical because we want those units to cancel out later)

51.88 inches
	1 inch

Step #3 Now consider where you want to go with this value. Since we want to go to metric, let's do a conversion here (Note: inches<=> centimeters is the only linear conversion you'll need to remember). Note: we read the second 'track' as "There are 2.54 centimeters in 1 inch)"

51.88 inches	2.54 cm
	1 inch

Step #4: Put the corresponding units from the top of the second track into the bottom of the third track along with the metric equivalent (Note: we read the third track as "In one meter there are 100 cm".

51.88 inches	2.54 cm	1 meter
	1 inch	100 cm

Step #5: Do the math:

51.88 x 2.54 x 1 ÷ 1 ÷ 100 = 1.318

and make sure to cancel out units that appear in BOTH the numerator and denominator. NOTE: Of *course* you don't have to multiply and divide by one, but please keep in mind I'm trying to teach METHOD, not math!

answer = 1.318 meters

This will take some time to learn. That's why we're gonna practice, practice, practice during the first several days. The GOOD news is we're going to learn about the physics of hurricanes (Atmospheric Physics, as it were) while we do!

 

1) The circumference of the Earth is about 25,000 miles around.

• Please convert that value into millimeters using railroad tracks
• Now please write your answer in scientific notation

Now let's mix it up a bit by including values in BOTH the numerator and denominator to start with. We'll start *gently*:

2) Usain Bolt ran the 100. meters at the 2009 World Track & Field Championships with a time of 9.58 seconds-- a world record that still stands.

Find his speed in miles/hr

3) Irma was one of the strongest hurricanes ever recorded when it hit the easter Caribbean last fall where maximum wind gusts were measured at 221.50 mph. Please convert that wind speed to meters/sec

ANSWER:

Here's a step-by-step on converting 221.50 mph into meters/sec

Let's do the linear measure first:

0) Write the given measurement in the first "track".

Notice that the top part of the track acts as the numerator and the bottom part of the track is the denominator.

Also, when we see the word 'per' that is another way of saying "divided by"

221.50 miles
1 hr

1a) Since we have miles in the top part of the first track we put miles in the bottom of the next track to make sure they will cancel each other out:

221.50 miles
1 hr	miles

1b) Then we ask ourselves if we want to go larger or smaller from miles in our conversion:

Since we're eventually trying to get to inches (so we can cross over to metric) we have to go smaller-- the next smallest English unit of measure that we use from miles is feet (we tend to ignore yards)

221.50 miles	feet
1 hr	miles

2) feet go to inches

Since we have feet on top, we put feet on bottom in the next track so they will cancel out.

We want to go smaller from feet to get to inches so we put inches on top in that track:

221.50 miles	feet	in
1 hr	miles	feet

3) inches go to centimeters (this is where we cross-over to metric)

Since we have inches on top, we put inches on bottom in the next track so they will cancel out.

We want to cross-over from English to Metric and we do that from cm to inches.... So we put cm on top in that track:

221.50 miles	feet	in	cm
1 hr	miles	feet	in

4) centimeters go to meters

Now that we're in metric we need to go bigger so we put meters on top

221.50 miles	feet	in	cm	m
1 hr	miles	feet	in	cm

We *think* we're done with the length conversions, but let's do our canceling first just to check. Every unit on the top row that has the corresponding unit on the bottom row will cancel out thusly (I've color-coded units to illustrate how nicely they all cancel out leaving us with METERS:

221.50 miles	feet	in	cm	m
1 hr	miles	feet	in	cm

5) Now that we're sure we've done the initial track-building correctly, let's fill in the numerical relationships in each track as follows:

• There are 5280 feet in one mile
• There are 12 inches in 1 foot
• There are 2.54 cm in 1 inch
• There are 100 cm in 1 m

221.50 miles	5280 feet	12 in	2.54 cm	1m
1 hr	1 miles	1 ft	1 in	100 cm

6) Now let's turn our attention to time!

Notice that time begins on the BOTTOM of the first track. We set that to cancel out with the same unit of time on the TOP of a NEW track.... after all, multiplication and division don't care what order the units are in. As long as one unit is on the top track and the same unit is somewhere else on the bottom of a track they will cancel!

221.50 miles	5280 feet	12 in	2.54 cm	1m	1 hr
1 hr	1 miles	1 ft	1 in	100 cm	60 min

7) minutes to seconds-- last step!

221.50 miles	5280 feet	12 in	2.54 cm	1m	1 hr	1 min
1 hr	1 miles	1 ft	1 in	100 cm	60 min	60 sec

8) Let's now check our units one last time and make sure everything cancels appropriately:

221.50 miles	5280 feet	12 in	2.54 cm	1m	1 hr	1 min
1 hr	1 miles	1 ft	1 in	100 cm	60 min	60 sec

We are left with meters on top and seconds on bottom (m/s) which is what we want!

Here's a VERY cool trick.

Many students are tempted to multiply all the top numbers and then write down the product in one final track--- and then multiply all the numbers on the bottom track and write THAT product in the bottom of that new track.

BOO!

Much too much can go wrong that way...

Mr W says:

Multiply all values on the top track and THEN divide by EACH value on the bottom track (ignoring all values of 1 of course!) thusly:

221.50 x 5280 x 12 x 2.54 ÷ 100 ÷ 60 ÷ 60 = 99.019 m/s YAY! (Don't forget sig figs. There were 5 s.f. in our measurement so there MUST be 5 sig figs in our answer. Don't worry about the sig figs in our conversion values, they don't count.

It takes a wee bit o' getting used to, but the chances of doing something DOH! go way, way down!

If time permits: *Gonzo* RR Trax: If I gave you the average number of gold atoms in a liter of ocean water, what information would need to have to calculate the value of ALL the gold in ALL the oceans of the world?