Representation of Fractions and Use of Constants
CHECK 6-5 Manipulate and solve for the following problem, using the information from Comprehension Checks 1–4
CHECK 6-5 Manipulate and solve for the following problem, using the information from Comprehension Checks 1–4.
Calculate the mass in kilograms of gravel stored in a rectangular bin 18.5 feet by 25.0 feet. The depth of the gravel bin is 15 feet, and the density of the gravel is 97 pound-mass per cubic foot.
LEARN TO: Recognize where and when to apply “reasonableness” within SOLVEM
EXAMPLE 6-1 Estimate how many miles of wire stock are needed to make 1 million standard paper clips.
Sketch:
See the adjacent diagram.
Objective:
Determine the amount of wire needed to manufacture a million paper clips.
Observations:
■ Paper clips come in a variety of sizes
■ There are four straight segments and three semicircular sections in one clip ■ The three semicircular sections have slightly different diameters
■ The four straight sections have slightly different lengths List of Variables and Constants:
■ L Overall length of clip ■ W Overall width of clip
■ L 1 , L 2 , L 3 , L 4 Lengths of four straight sections L
W
Erroneous or argumentative thinking can lead to problem-solving errors. For example, ■ I can probably find a good equation in the next few pages. Perhaps you read a
prob-lem and rifle through the chapter to find the proper equation so that you can start substituting numbers. You find one that looks good. You do not worry about wheth-er the equation is the right one or whethwheth-er the assumptions you made in committing to the equation apply to the present problem. You whip out your calculator and produce an answer. Don’t do this!
■ I hate algebra, or I cannot do algebra, or I have got the numbers, so let us substitute the values right in. Many problems become much simpler if you are willing to do a little algebra before trying to find a numerical solution. Also, by doing some manip-ulation first, you often obtain a general expression that is easy to apply to another problem when a variable is given a new value. By doing a little algebra, you can also often circumvent problems with different sets of units. Do some algebra!
■ It is a simple problem, so why do I need a sketch? Even if you have a photographic memory, you will need to communicate with people who do not. It is usually much simpler to sort out the various parts of a problem if a picture is staring you right in the face. Draw pictures!
■ I do not have time to think about the problem, I need to get this stuff finished. Well, most often, if you take a deep breath and jot down several important aspects of the problem, you will find the problem much easier to solve and will solve it correctly.
Take your time!
6.4 EXAMPLES OF SOLVEM
6.3 AVOIDING COMMON MISTAKES
LEARN TO: Adopt strategies that will assist in reducing errors in problem solving.
■ D 1 , D 2 , D 3 Diameters of three semicircular sections ■ P 1 , P 2 , P 3 Lengths of three semicircular sections ■ A Total amount (length) of wire per clip Estimations and Assumptions:
■ Length of clip: L = 1.5 in ■ Width of clip: W = 3>8in
■ Diameters from largest to smallest • D1 = W = 3>8in
• D2 = 5>6in • D3 = 1>4in
■ Lengths from left to right in sketch • L1 = To be calculated
• L2 = 0.8 in • L3 ⬇ L4 = 1 in
Equations:
■ Perimeter of semicircle: P = pD > 2 (half of circumference of circle) ■ L1 = L - D1>2 - D2>2
■ Total length of wire in clip: A= L1 + L2 + L3 + L4 + P1 + P2 + P3 Manipulation:
In this case, none of the equations need to be manipulated into another form.
Length of longest straight side: L1 = 1.5 - 3>16 - 5>32 ⬇ 1.2 in Lengths of semicircular sections: P1 = p 3>16 ⬇ 0.6 in
P2 = p 5>32 ⬇ 0.5 in P3 = p 1>8 ⬇ 0.4 in
Overall length for one clip: A= 1.2 + 0.8 + 1 + 1 + 0.6 + 0.5 + 0.4 = 5.5 in/clip
Length of wire for 1 million clips: 15.5 in> clip2 11 * 106clips2 = 5.5 * 106 in Convert from inches to miles: 15.5 * 106 in2 11 ft> 12 in2
11 mile> 5,280 ft2 ⬇ 86.8 miles One million, 1.5-inch paper clips require about 87 miles of wire stock.
EXAMPLE 6-2 A spherical balloon has an initial radius of 5 inches. Air is pumped in at a rate of 10 cubic inches per second, and the balloon expands. Assuming that the pressure and temperature of the air in the balloon remain constant, how long will it take for the surface area to reach 1,000 square inches?
Sketch:
Rf Ri
Objective:
Determine how long it will take for the surface area of the balloon to reach 1,000 in 2
Observations:
■ The balloon is spherical
■ The balloon, thus its volume and surface area, gets larger as more air is pumped in ■ The faster air is pumped in, the more rapidly the balloon expands
List of Variables and Constants:
■ Initial radius: Ri = 5 [in]
■ Final radius: R f [in]
■ Initial surface area: A i [in 2 ] ■ Final surface area: A f [in 2 ] ■ Change in volume: ΔV [in 3 ] ■ Initial volume: V i [in 3 ] ■ Final volume: V f [in 3 ] ■ Fill rate: Q = 10 [in3/s]
■ Time since initial size: t [s]
Equations:
■ Surface area of sphere: A = 4pR2 ■ Volume of sphere: V = 4/3 pR3 ■ Change in volume: V = Qt Manipulation:
There are a few different ways to proceed. The plan used here is to determine how much the balloon volume changes as air is blown into the balloon and to equate this to an expression for the volume change in terms of the balloon geometry (actually the radius of the balloon).
Radius of balloon in terms of surface area: R = a A 4pb1>2 Final balloon radius in terms of surface area: Rf = aAf
4pb1>2 Final volume of balloon in terms of surface area: Vf = a4p
3 b aAf 4pb3>2 Volume change in terms of air blown in: V = Vf - Vi = Qt Volume change in terms of geometry: Vf - Vi = a4p
3 b aAf
4pb3>2 - a4p 3 bR3i Solve for time to blow up balloon: t = a4p
3Qb aAf
4pb3>2 - a4p 3QbR3i
And simplifying: t = a4p
3Qb e aAf
4pb3/2- R3if It takes just over 4 minutes to increase the volume to 1,000 cubic inches.
NOTE
Whenever you obtain a result in equation form, you should check to see if the dimensions match in each term.
IMPORTANT
Recall that you should manipulate the equations before inserting known values. Note that the final expression for elapsed time is given in terms of initial radius (R i ), flow rate (Q), and final surface area (A f ). So you now have a general equation that can be solved for any values of these three parameters. If you had begun substituting numbers into equa-tions at the beginning and then wanted to obtain the same result for different starting values, you would have to resolve the entire problem.
In-Class Activities
ICA 6-1
Each of these items should be addressed by a team. List as many things about the specified items as you can determine by observation. One way to do this is to let each team member make one observation, write it down, and iteratively canvass the team until nobody can think of any more additions (it is fine to pass). Remember that you have five senses. Also note that observations are things you can actually detect during the experiment, not things you already know or deduce that cannot be observed.
To help, Examples A and B are given here before you do one activity (or more) on your own. Remember that not all observations will be important for a particular problem, but write them down anyway—they may trigger an observation that is important.
Example A: A loudspeaker reproducing music
■ Electrical signal is sent to speaker.
■ Speaker vibrates from electrical signal.
■ Speaker diaphragm is made of paper;
vibrating air creates sound.
■ Friction between straw and lips allows you to hold it.
Example B: Drinking a soft drink in a can through a straw
From the list below or others as selected by the instructor, list as many observations as you can about the following topic:
(i) Large container guided smoothly up a ramp (j) Ruler hanging over the desk
(k) Your computer (when turned off) (l) Your chair
(m) Your classroom
(n) A weight tied on a string and twirled
Final Assignment of this ICA: You have done several observation exercises. In these, you thought of observation as just a “stream of consciousness” with no regard to organization of your efforts. With your previous observation as a basis, generalize the search for observations into several (three to six or so) categories. The use of these categories should make the construction of a list of observations easier in the future.
Analyze the following problems using the SOLVEM method.
ICA 6-2
What diameter will produce a maximum discharge velocity of a liquid through an orifice on the side at the bottom of the cylindrical container? Consider diameters ranging from 0.2 to 2 meters.
ICA 6-3
A hungry bookworm bores through a complete set of encyclopedias consisting of n volumes stacked in numerical order on a library shelf. The bookworm starts inside the front cover of volume 1, bores from page 1 of volume 1 to the last page of the last volume, and stops inside the back cover of the last volume. Note that the book worm starts inside the front cover of volume 1 and ends inside the back cover of volume n .
Assume that each volume has the same number of pages. For each book, assume that you know how thick the cover is, and that the thickness of a front cover is equal to the thickness of a back cover; assume also that you know the total thickness of all the pages in the book. How far does the bookworm travel? How far will it travel if there are 13 volumes in the set and each book has 2 inches of pages and a 1>8 -inch thick cover?
ICA 6-4
Two cargo trains each leave their respective stations at 1:00 p.m. and approach each other, one traveling west at 10 miles per hour and the other on separate tracks traveling east at 15 miles per hour. The stations are 100 miles apart. Find the time when the trains meet and determine how far the eastbound train has traveled.
ICA 6-5
Water drips from a faucet at the rate of three drops per second. What distance separates one drop from the following drop 0.65 seconds after the leading drop leaves the faucet? How much time elapses between impacts of the two drops if they fall onto a surface that is 6 feet below the lip of the faucet?
Your sketch should include the faucet, the two water drops of interest, and the impact sur-face. Annotate the sketch, labeling the each item shown and denote the relevant distances in symbolic form, for example, you might use d 1 to represent the distance from the faucet to the first drop.
ICA 6-6
During rush hour, cars back up when the traffic signal turns red. When cars line up at a traffic signal, assume that they are equally spaced (≤x) and that all the cars are the same length (L).
You do not begin to move until the car in front of you begins to move, creating a reaction time (≤t) between the time the car in front begins to move and the time you start moving. To keep things simple, assume that when you start to move, you immediately move at a constant speed (v).
(a) If the traffic signal stays green for some time (t g ), how many cars (N) will make it through the light?
(b) If the light remains green for twice the time, how many more cars will get through the light?
(c) If the speed of each car is doubled when it begins to move, will twice as many cars get through the light? If not, what variable would have to go to zero for this to be true?
(d) For a reaction time of zero and no space between cars, find an expression for the number of cars that will pass through the light. Does this make sense?
ICA 6-7
Suppose that the earth were a smooth sphere and you could wrap a 25,000-mile-long band snugly around it. Now let us say that you lengthen the band by 10 feet, loosening it just a little.
What would be the largest thing that could slither under the new band (assume that it is now raised above the earth’s surface equally all the way around so that it doesn’t touch anywhere):
an amoeba, a snake, or an alligator?
Chapter 6 REVIEW QUESTIONS
Analyze the following problems using the SOLVEM method.
1. A motorcycle weighing 500 pounds-mass plus a rider weighing 300 pounds-mass produces the following chart. Predict a similar table if a 50-pound-mass dog is added as a passenger.
Velocity (v) [mi/h] Time ( t ) [s]
0 0.0
10 2.3
20 4.6
30 6.9
40 9.2
2. A circus performer jumps from a platform onto one end of a seesaw, while his or her partner, a child of age 12, stands on the other end. How high will the child “fly”?
3. Your college quadrangle is 85 meters long and 66 meters wide. When you are late for class, you can walk (well, run) at 7 miles per hour. You are at one corner of the quad and your class is at the directly opposite corner. How much time can you save by cutting across the quad rather than walking around the edge?
4. I am standing on the upper deck of the football stadium. I have an egg in my hand. I am going to drop it and you are going to try to catch it. You are standing on the ground. Apparently, you do not want to stand directly under me; in fact, you would like to stand as far to one side as you can so that if I accidentally release it, it won’t hit you on the head. If you can run at 20 feet per second and I am at a height of 100 feet, how far away can you stand and still catch the egg if you start running when I let go?
5. A 1-kilogram mass has just been dropped from the roof of a building. I need to catch it after it has fallen exactly 100 meters. If I weigh 80 kilograms and start running at 7 meters per second as soon as the object is released, how far away can I stand and still catch the object?
6. Neglect the weight of the drum in the following problem. A sealed cylindrical drum has a diameter of 6 feet and a length of 12 feet. The drum is filled exactly half-full of a liquid having a density of 90 pound-mass per cubic foot. It is resting on its side at the bottom of a 10-foot deep drainage channel that is empty. Suppose a flash flood suddenly raises the water level in the channel to a depth of 10 feet. Determine if the drum will float. The density of water is 62 pound-mass per cubic foot.
LEARNING OBJECTIVES
The overall learning objectives for this unit include:
Chapter 7:
■ Identify basic and derived dimensions and units.
■ Express observations in appropriate units and perform conversions when necessary. Apply the laws governing equation development to aid in problem solutions.
Chapter 8:
■ Apply basic principles from mathematical and physical sciences, such as the conservation of energy and the ideal gas law, to analyze engi-neering problems.
■ Convert units for physical and chemical parameters such as density, energy, pressure, and power as required for different systems of units.
■ Use dimensions and units to aid in the solution of complex problems.
Chapter 9:
■ Identify when a quantity is dimensionless.
■ Using a graph of dimensionless groups, extract information from the plot about the physical system.
■ Given a set of parameters, determine appropriate dimensionless groups using Rayleigh’s Method.
■ Determine the Reynolds Number; interpret the Reynolds Number for fluid flow in a pipe.
Ubiquitous: yoo·bik·we·teous ~ adjective;
definition __________________________________________________________
Part 2 UBIQUITOUS UNITS
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