Warm Up – Dimensional Analysis Practice Susanna is 5.50 ft tall. What is her height in centimeters? (1 in = 2.54 cm) 2. If we are in class for 1.5 hours, how many seconds are we in class?

Warm Up – Dimensional Analysis Practice We are going to plan a hypothetical pizza party. How much will it cost? (write it down with explanation) If you’re stuck consider these questions: How many students are in the class? How many slices of pizza do you think each student will eat? How many slices of pizza are in a whole pizza How much does a whole pizza cost?

Let’s Throw a Pizza Party How much will it cost for our class? How many students? Slices per student? Slices per pizza? Price per pizza? Work out the problem on the board, either white, or smartboard if you have one

How to convert units? (1) So What?: You can’t add euros to dollars You can’t add feet to feet/sec Step 1: When to convert Numbers with units, like 16.2 meters or 32 ft/sec², are treated exactly the same as coefficients with variables, like 16.2x or 32y/z². Once you grasp this, you see at once why the laws of units work as they do. You can’t add 32 ft to 32 ft/sec, any more than you can add 32x to 32x/y. And when you divide 32 miles by 4 hours to get 8 miles/hour, that’s exactly the same as dividing 32x by 4y to get 8x/y.

How to convert units? (2) Step 2: Multiply by 1 You can multiply anything by 1 and not change its value. Convert 4.5 hours to minutes: (1) 60 minutes = 1 hour (2) 60 minutes = 1 1 hour (3) 4.5 hr × 60 min 1 hr (4) 4.5 hr x 60 min = 4.5x 60 min =

Is it really multiplying by 1? In which case will you overcook the Turkey? Which is longer, cooking a Turkey for 3.25 hours or 195 minutes? Note: If you converted 3.5 hours and came out with .0583 minutes, you probably messed up. Always see if the answer makes sense But wait a minute!” I hear you say. “You started with 4.5 and ended up with 270. How is that multiplying by 1?” The answer is that we didn’t start with a “dimensionless” pure number 4.5, but with 4.5 hours; and we didn’t end up with a pure number 270 but with 270 minutes. You should be able to convince yourself that if you bake a turkey for 270 minutes or 4½ hours, either way you wait the same length of time for dinner. A number with units is different from a number without units or with different units, just as 8x is different from both 8 and 8y. Think of it this way: 3.27 dollars or euros is the same as 327 cents, when you multiply by the “carefully chosen form of 1”, 100 cents/dollar or 100 cents/euro. If the top and bottom of the fraction are equal, the fraction equals 1 and the value after multiplying is the same as the value before multiplying — but expressed in different units, which of course is the whole point. You might be asking yourself, “Why all the fuss? Anybody knows that to convert hours to minutes you have to multiply by 60.” Well, yes, that’s true. But I deliberately chose a simple example to show the method. I’ll try to use more realistic (i.e., harder) examples from here on.

How to convert units? (3) Step 3: How to pick a “1” Find a conversion factor between the given units and the desired units, and write it as an equation. EX: to convert km to miles 1 km = 0.621 miles (2) Convert that equation to a fraction with the desired units on top and the given units on the bottom 1 km  there’s your “1” conversion 0.621 miles factor

How to convert units? (4) Step 4: Chaining Conversions  combine conversions to avoid looking up a specific conversion Like in the pizza example EX: how many meters are in the 440-yard dash? 440 yd = 402 m

We start by writing down the Converting Inches to centimeters 10.0 in We start by writing down the number and the unit

Converting Inches to centimeters 10.0 in 2.54 cm 1 in Our conversion factor for this is 1 in = 2.54 cm. Since we want to convert to cm, it goes on the top.

Converting Inches to centimeters 10.0 in 2.54 cm 1 in Now we cancel and collect units. The inches cancel out, leaving us with cm – the unit we are converting to.

Converting Inches to centimeters 10.0 in 2.54 cm 25.4 cm = 1 in The Since the unit is correct, all that is left to do is the arithmetic... The Answer

Recap of the Procedure Find conversion factor as a fraction with the given units in the opposite position from the original measurement Raise the conversion fraction to that same power Multiply the original measurement by the conversion fraction, and simplify.