Download PISTON LAB

Pistons and Trig. Functions
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Tom Berg
Mitchell High School
920 N. Capital, Mitchell, SD 57301
email: [email protected]
phone: (605) 995-3034
Integrated Math II, Grades 10-12
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Class setup: Students will be divided into heterogeneous groups of 3 to 4 students.
Teacher’s role: The teacher’s role is to present the lab to the students. While the students are
working on the lab, the teacher will be walking around the room monitoring the students.
Textbook and other resources: CORD Applied Mathematics (Center for Occupational
Research and Development), Waco, Texas (1994), Unit 22-Using Trigonometric Functions.
Content/topic/theme: Trigonometric Functions
Real-world experience: I don’t believe that this will give them a real-world experience but
rather a real hands-on experience. As students rotate the piston, they will see how the wave is
produced and how to find the amplitude, frequency, and period. Hopefully, they will relate the
modeling of this wave to other waveforms.
Why do students need to know this? Students will learn about sinusoidal waves and their
connection to sound waves, vibration waves, electrical waves (AC current), ultrasonic waves,
and microwaves.
What is the connection to prior learning experience ? Students will use skills previously
taught in Unit 1: Learning Problem-Solving Techniques, Unit 2: Estimating Answers, Unit 6:
Working with Lines and Angles, Unit 9: Using Ratios and Proportions, Unit 11: Using Signed
Numbers and Vectors, Unit 15: Using Formulas to Solve Problems, Unit 17: Graphing Data and
Unit 21: Using Right-Triangle Relations.
Discipline-specific standard(s):
National Math Standards:
Algebra: Understand Patterns and Use Mathematical Models
Problem Solving
Connections
Representation
South Dakota State Math Standards:
Goal 1: Algebra;
Indicator 1: Analyze procedures to transform algebraic
expressions;
Standard 7: extend the concepts of algebra to other types of
functions, e.g., trigonometric, exponential, and
logarithmic
Goal 5: Patterns, Relations, and Functions;
Indicator 1: Analyze and describe the properties and behaviors of
relations, functions, and related inverses;
Standard 1: use various representations of functions, e.g., graphs,
tables, symbolic forms.
Goal 5: Patterns, Relations, and Functions;
Indicator 1: Analyze and describe the properties and behaviors of
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relations, functions, and related inverses;
Standard 8: create tables or graphs to interpret relations and/or
functions.
Goal 5: Patterns, Relations, and Functions;
Indicator 2: Apply relations and functions to complex problem-solving
situations;
Standard 5: identify natural phenomena that are cyclic.
Activities (What will the students do?) The students will do a hands-on math lab involving a
piston and relating this to a cosine wave.
Strategy (How will they do it?) The students will have a model of a piston. The piston will be
in its tallest position (top-dead-center). The students will then measure how far the top of the
piston is from the base (height). The students will turn the piston in 30° increments and take the
height measurements. They will continue turning the piston until two revolutions (720°) have
been achieved. The students will make graphs from their tables of values and answer the
questions. The students will also use the graphing calculator to plot the points. They will also
type in their equations and see if their graphed equations match their plotted points.
Assessment strategies: The students will be assessed on how they answer the questions in the
lab. The students need to fill in the table of values; make graphs (including labeling of the x and
y axes); answering questions about the kind of curve it is, the amplitude, the period, the
frequency, and the vertical shift. They are also graded on the correct placement of these items
into the equations.
Warm-up or closure strategy: At the beginning of the class, the students will be given a graph
of a wave. The students must find the amplitude, period, and frequency of this wave. They must
also identify whether it is a sine wave or a cosine wave.
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From Unit 22 - Using Trigonometric Function
Exercise 20: During the stroke of a piston, the connecting rod pushes the crankshaft
around 360°, as illustrated here.
As the crankshaft turns, the radius of the "throw" makes an angle T with a line between the shaft
and the end of the rod (as shown in the drawing). The changing distance between the end of the
rod and the crankshaft center can be computed using the formula below. (This changing distance
can be used to determine the volume of gas/air mixture ahead of the piston as the crankshaft
rotates.)
where h is the distance between the end of the rod and the crankshaft center,
r is the radius from the crankshaft center to the throw,
d is the length of the rod, and
T is the angle made by the radius r and a line between the crankshaft center and the end
of the rod.
a. Using a value of r = 2.5 inches and d = 6.0 inches, make a table of values for h for angles of T
equal to 0°, 30°, 60°, and so on, up to 360°.
b. Draw a graph of these values for d, and connect the plotted points with a smooth curve.
c. Referring to the drawing, you should be able to predict the value for d when T = 0° and
T = 180°. Check the results from the formula with your expectations.
d. Examine the shape of the curve in your graph. Which curve does it most resemble—a sine
curve or a cosine curve?
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PISTON LAB
Start with the piston in the top-dead-center (TDC) position. This is the 0° position. Measure the
height of the piston head from the base. Turn the gear 30° clockwise, measure the height.
Continue turning the gear 30° for 2 revolutions and take the height readings (in cm).
a. Fill in the chart and graph the results.
Gear
angle
height
0
30
60
90
120
150
180
210
240
270
300
330
360
390
420
450
480
510
540
570
600
630
660
690
720
b. Does this look like a sin or cos curve?
c. What is the amplitude?
d. What is the period?
e. What is the frequency?
f. What is the vertical shift?
g. Write the equation: ht =
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