FLUID STATION 1.1 MATH MODULE UNDERSTANDING PRESSURE STEMPrep Project 2015 Unlimited Learning, Inc. Ciatlyn Reese, Author

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PRESSURE What is pressure? Surely you’ve heard someone say they’re “under pressure,” meaning they’re being persuaded or forced to do something. But in science, the definition of pressure is … The force that a substance or object puts on the surface of something else A perfect example would be if you are pushing your hands against an object, like a wall. You are putting pressure on the wall by using your hands to apply force to its surface.

BUT PRESSURE ISN’T ONLY FOR SOLID OBJECTS! Fluids, which include both liquids and gases, can also apply pressure to other things, and have pressure applied to them! For example… Water puts pressure on the inside surface of a glass… (And the glass also puts pressure on the water!) And air puts pressure on the inside surface of a balloon… (And the balloon also puts pressure on the air!)

THE PRESSURE EQUATION Actually, there is an equation we can use to calculate this: So now we have an idea of what pressure is. But how do we figure out the amount of pressure being put on something? And you may see this equation written as simply: P = F/A *But what do we actually mean by force? How is it measured?

FORCE What is force? In science, a force is the push or pull of one object on another object (or substance) There are actually 2 types of forces: Contact forces, which happen when two objects (or substances) physically touch and Action-at-a-distance forces, which can happen even when two objects are very far away from each other!  (A good example is gravity. The sun exerts a force on the Earth to keep it in orbit, even though they are separated by millions of miles!)

UNITS OF FORCE Force is measured in a strange unit that you may have never heard of, called a Newton, named after Isaac Newton who wrote the 3 laws of motion! (You have probably heard the story about an apple falling on his head, and how Newton then discovered gravity.) But you don’t have to worry about all that! Just remember that force is measured in Newtons.

But what good is a Newton when we don’t even really understand how much force 1 Newton is? Let’s look at some examples: Pssst! The abbreviation for Newtons is “N”! A person who weighs 150 pounds puts a force of 667 N on the Earth A stick of margarine puts a force of about 1 N on a counter top A 2 ton car that is speeding up and gaining 5 mph every second has a force of 4,055 N!

I’ll bet that you’ve probably heard of and even done math with area before. But just in case, let’s review it. The definition of area is… The amount of space inside the boundary of a two-dimensional (flat) surface We can find the area of a flat object, such as a piece of paper… We can find the area of a flat surface, on a three dimensional object, such as a box… OR Area = 88 in 2 Front Surface = 32 in 2 PSSST! The standard unit of area (most commonly used all over the world) is square meters (m 2 ), but there are many other units that are also used, such as square centimeters (cm 2 ), square inches (in 2 ) and square miles (mi 2 ).

We can use several equations to find the areas of different shapes, but the one you are probably the most familiar with is for rectangles and squares: Let’s look at an example: Length = 12 in Height = 8 in LengthHeight x = AREA NOTICE THE UNITS! Since we are multiplying inches x inches, the final unit is inches squared (in 2 ). This is true for all units of area! AREA OF A RECTANGLE

Let’s look at an example: LengthHeight x = AREA AREA OF A PARALLELOGRAM Parallelograms are kind of like rectangles that have been tilted to the side. So, the formula for calculating area is the same: Watch out! “Height” means a straight line from top to bottom, NOT the length of the side! Length = 8 in Height = 7 in

3 cm 10 cm Now try it yourself! Find the areas of the shapes below, and write your answers on the “Lesson Practice Problems” print out. When you are done, click to check your work. 5.5 in 6 in 4 in

3 cm 10 cm 5.5 in 6 in 4 in Check your answers. Cross out any wrong answers and write the correct answers on your answer sheet.

The area of a triangle has a slightly different equation for finding area: AREA OF A TRIANGLE IMPORTANT! “Base” simply means the length along the bottom of the triangle. Again, “Height” means a straight line from top to bottom; this is NOT always the same as the length of a triangle’s side! In a right triangle, the height is the same as the length of the side because the right angle makes a perfectly vertical (straight up-and-down) side!! But in all other triangles, you must look for a dotted line down through the middle which shows you how much the actual height is! Height = 4 cm Base = 5 cm Base = 3 cm Height = 4 cm

An easy way to understand this is to remember that any triangle is really just ½ of a square, rectangle or parallelogram! Equal But why do we divide it by two? Length Height

BaseHeight x = AREA Height = 8 cm Base = 5 cm ÷ 2 BaseHeight x = AREA ÷ 2 This is not a right triangle! Look for the dotted line… This is a right triangle! Look for the height on the side. Base = 10 in Height = 2 in Here are some examples:

Now try to find the areas of the triangles below, and write your answers on the “Lesson Practice Problems” print out. When you are done, click to check your work. 13 in 2 in 8 cm 1cm 4 in 7 in 5 cm 12 cm

13 in 2 in 8 cm 1cm 4 in 7 in 5 cm 12 cm Check your answers. Cross out any wrong answers and write the correct answers on your answer sheet.

The equation for finding the area of a circle is the most unlike the others: AREA OF A CIRCLE First lets define some common parts of a circle…

radius The “radius” is the distance from the center to any edge of the circle diameter The “diameter” is the distance through the center, across the entire circle The “circumference” is the distance all the way around the edge of the circle

And finally… You may have heard of pi before (pronounced “pie.”) Pi is a number with a long, crazy decimal at the end that goes on forever and never repeats. But in most cases, we simply round pi to 3.14 in order to use it in math problems. For ANY circle, the circumference (C) divided by the diameter (d) equals pi ÷ That’s right! No matter how big the circle, it always equals pi! Check it out… C = 15.7 in ÷ d = 5 in C = 22 in ÷ d = 7 in

Okay! Now that we know what π and a radius are, we can figure out the area of a circle using our equation: Here’s an example: π (Radius) 2 x AREA Radius = 5 cm

Radius = 3 cm Radius = 6 in Radius = 1 cm Radius = 10 cm Now try to find the areas of the circles below, and write your answers on the “Lesson Practice Problems” print out. When you are done, click to check your work.

Check your answers. Cross out any wrong answers and write the correct answers on your answer sheet.

PRESSURE Remember, in math and science pressure is defined as: The force that a substance or object puts on a surface of something else which uses the equation: So what do you think the units of pressure are? Well, there are actually lots of units for expressing pressure! Memory Jogger: Newtons are the standard unit for FORCE and meters are the standard unit for AREA

Now let’s try calculating pressure with some example problems: Let’s say a box is sitting on a table top. If the box is putting a force of 10 Newtons on the table, and the side of the box that is touching the table has an area of 0.25 square meters, how much pressure is being put on the table? In this problem, we can simply plug the numbers straight into our equation: Force ÷ Area = Pressure That was almost too easy! Let’s try another one…

Some gas is contained in a cylinder with a piston. The gas is pushing upwards on the bottom of the piston with a force of 150 N. The radius of the piston is 0.4 m. How much pressure is the gas putting on the piston? A little harder… This time we are not given the area! We are going to have to find it using our equation for calculating area of a circle (since the piston is round), which is Area = π x (radius) 2 Radius = 0.4 m Bottom of the piston: Now we can solve for pressure (P = F/A): (Round to 1 decimal point)

Okay! Ready to try it? Find the pressures in the two problems below, and write your answers on your answer sheet. When you are done, click to check your work. A triangular pyramid is sitting on a table top. The pyramid is putting a force of 5 Newtons on the table. The bottom of the pyramid that is touching the table has base of 1m, and height of 0.5 m. How much pressure is being put on the table? (Hint: You will have to find the area of the pyramid’s base first). Base of the pyramid: Height = 0.5 m Base = 1 m

A triangular pyramid is sitting on a table top. The pyramid is putting a force of 5 Newtons on the table. The bottom of the pyramid that is touching the table has base of 1m, and height of 0.5 m. How much pressure is being put on the table? (Hint: You will have to find the area of the pyramid’s base first). Base of the pyramid: Height = 0.5 m Base = 1 m Check if you got the correct answer. If not, cross it out and write the correct answer on your answer sheet. Now we can solve for pressure (P = F/A):

Some gas is contained in a cylinder with a piston. The gas is pushing upwards on the bottom of the piston with a force of 200 N. The radius of the piston is 0.08 m. How much pressure is the gas putting on the piston? (Hint: You will need to find the area of circular piston first.) Practice Problem #2 Radius = 0.08 m Bottom of the piston:

Some gas is contained in a cylinder with a piston. The gas is pushing upwards on the bottom of the piston with a force of 200 N. The radius of the piston is 0.08 m. How much pressure is the gas putting on the piston? (Hint: You will need to find the area of the circle on the bottom of the piston first.) Radius = 0.08 m Bottom of the piston: Now we can solve for pressure (P = F/A): Check if you got the correct answer. If not, cross it out and write the correct answer on your answer sheet. (Round to 1 decimal point)

OTHER UNITS OF PRESSURE Like we said before, there are lots of different units for pressure; but we’re just going to work with the ones you’re most likely to see:

PASCALS You’re probably wondering, well what happened to Newtons per square meter (N/m 2 )? Actually, N/m 2 and Pascals (Pa) are the same thing! This unit is named after Blaise Pascal who was a respected French scientist from the 1600’s. Pascal made many significant contributions to the understanding of pressure. l#/media/File:Blaise_Pascal_Versailles.JPG As mentioned before, N/m 2 or Pascals (Pa) are the standard unit of pressure across the world, so they are very commonly used:

ATMOSPHERES Another very common unit of pressure is atmospheres (atm). You may have even heard of them before: When on the Earth’s surface, the amount of pressure we feel from the surrounding air has been defined as 1 atmosphere (1 atm). This pressure then decreases gradually as your altitude (distance from the Earth’s surface) increases. In space, there is no air, and therefore no pressure (0 atm). So how do atmospheres compare to Pascals? Well… WOAH! Sounds like a lot, right? Actually, we don’t even feel this pressure because we are used to it!

MILLIMETERS OF MERCURY Finally! We have reached the unit of pressure that you saw in the lab activity – millimeters of mercury (mmHg): The mercury manometer was the first pressure measuring device. A glass tube was filled with liquid mercury and excess pressure (pressure greater than that of the atmosphere) caused the level of the mercury to rise up the tube. The distance it rose was measured in millimeters. Since this invention, more accurate devices have been invented for measuring pressure. But, millimeters of mercury (mmHg) are still used in medicine (such as measuring blood pressure). So how do mmHg compare to Pascals and atmospheres? Well… Modern mercury manometer g/wiki/File:Mercury_manomet er.jpg

PRESSURE CONVERSIONS Let’s start with converting between Pascals (Pa) and atmospheres: Remember we said that: This is called a conversion factor. If we know the conversion factor for two units, we can convert easily in either direction! We can set up this up in two different ways: OR *The one we choose will depend on which direction we are converting. We pick the one that will cause the units we DON’T want to cancel out! This may be confusing, so let’s look at an example…

PASCALS AND ATMOSPHERES Example: Convert 3 atm to Pa We ALWAYS multiply by the conversion factor But which one do we use? Well, we want atm to cancel, (leaving Pa) so we need the one where atm is on bottom This is because units cancel just like numbers! atm = 1 So if we put atms on bottom, they basically go away! Now we’re left with: Multiply Then divide

PASCALS AND ATMOSPHERES We can use this strategy for ALL unit conversion problems! All we need is a conversion factor! Let’s try the other direction this time… This time, we want Pa to cancel, so we set up the conversion factor with Pa on bottom Now we’re left with: Multiply Then divide

Whew! Ready to try it? Try the two problems below, and write your answers on the “Lesson Practice Problems” print out. and then click to check your work.

Check your answers. Cross out any wrong answers and write the correct answers on your answer sheet.

PRESSURE CONVERSIONS Now let’s try converting between atmospheres (atm) and millimeters of mercury (mmHg) Remember we said that: We can do these conversions in exactly the same way we did on the last few slides! Here are our two possible conversion factors that we can get from the information above: OR Ready? Let’s do this!

ATMOSPHERES AND MILLIMETERS OF MERCURY Example: Convert 2.5 atm to mmHg Multiply by the conversion factor Now we’re left with: Multiply Then divide Right now we want atm to cancel, so we set up the conversion factor so that atm is on bottom

ATMOSPHERES AND MILLIMETERS OF MERCURY Example: Convert 3800 mmHg to atm Multiply by the conversion factor Now we’re left with: Multiply Then divide Now we want mmHg to cancel, so we set up the conversion factor so that mmHg is on bottom

Okay! Let’s try it! Try the two problems below, and write your answers on the “Lesson Practice Problems” print out. Then click to check your work.

Check your answers. Cross out any wrong answers and write the correct answers on your answer sheet.

PRESSURE CONVERSIONS Okay, ready for an even bigger challenge? Let’s convert Pascals all the way to mmHg! Remember:

PASCALS, ATMOSPHERES AND MILLIMETERS OF MERCURY Example: Convert 60,795 Pa to mmHg Now we’re left with: Multiply Then divide Notice how first we convert to atm But with the second conversion factor we cancel out atm and are left with mmHg

PASCALS, ATMOSPHERES AND MILLIMETERS OF MERCURY Example: Convert 304 mmHg to Pa Now we’re left with: Multiply Then divide Again, first we convert to atm And the second conversion factor cancels out atm leaving only mmHg Okay, now let’s try it the other way

Whew! Okay, let’s try it! Try the two problems below, and write your answers on the “Lesson Practice Problems” print out. Then and then click to check your work.

Check your answers. Cross out any wrong answers and write the correct answers on your answer sheet. Check it out! That’s a perfect systolic (top) blood pressure! And here’s a great diastolic (bottom) blood pressure!

REVIEW AND PRACTICE TURN IN YOUR FIRST ANSWER SHEET TO YOUR COACH. THEN COMPLETE THE FOLLOWING WORKSHEETS (YOU SHOULD HAVE A PRINT OUT OF THEM): FINDING AREA OF MIXED SHAPES CALCULATING PRESSURE (IN N/M 2 ) PRESSURE CONVERSIONS ONCE YOU COMPLETE EACH WORKSHEET, ASK YOUR COACH FOR THE ANSWER KEY AND CORRECT YOUR WORK. DON’T WORRY, YOU’LL ONLY BE GRADED FOR COMPLETION ON THIS PART. FINALLY, CLICK HERE TO REVIEW SOME FLASH CARDS WITH ALL OF THE TERMS YOU LEARNED IN THIS LESSON (YOU CAN ALSO PLAY GAMES WITH THE TERMS!):

CONGRATULATIONS! YOU NOW HAVE A GOOD INTRODUCTION TO CALCULATING PRESSURE AND UNIT CONVERSIONS! Ready to take the quiz?