Rope Tension: Rate of Change
Grade Level:
Duration:
11/12
1 Day
Subject:
Pre-Calculus
Prepared By:
Nick Hanlon
Day 1
Materials Needed
Plywood board
Hooks
Rope
Any object with mass
Protractors
Analyze Learners
Overview & Purpose (STEMcinnati theme)
Education Standards Addressed
The purpose of the lesson is to illustrate the rate of change in rope tension. The
example is based on any object in equilibrium, where an object is hung with two
ropes supporting the weight, such as a traffic light. Students should be familiar
with force triangles and Law of Sines. This lesson applies the force triangles and
Law of Sines to show the rate of change in the rope tension as the angle between
the rope and horizon decreases.
(Math) Number, Number Sense and Operations
Standard
Use vector addition and scalar multiplication to solve
problems.
Overview:
A: Cranes, human muscles and joints, banners, signs
C: Biomedical, Civil and Materials Engineering (see description of each field at the
end of the lesson plan)
S: Ability to safely move large items with minimal stress
(Math) Measurement Standard
Use radian and degree angle measures to solve problems
and perform conversions as needed.
(Math) Geometry and Spatial Sense Standard
Use trigonometric relationships to determine lengths and
angle measures; i.e., Law of Sines and Law of Cosines.
(Science) Physical Sciences Standard
D. Apply principles of forces and motion to mathematically
analyze, describe and predict the net effects on objects or
systems.
Select Goals and
Objectives
Teacher Guide
Student Guide
Assessment
Goals and
Objectives
(Specify
skills/information that
will be learned.)
Goals:
1. Students should understand how to draw a force
triangle
2. Students should understand how to find the
magnitude of a vector of the force triangle.
3. Students should understand the rate of change in the
rope tension as the angle from the horizon to the rope
decreases.
Objectives:
1. Students will be able to draw a force triangle from a
free-body-diagram and label the appropriate angles.
2. Students will be able to calculate the magnitude of a
vector by using Law of Sines.
3. Students will be able to graph the relationship
between angle and tension.
Formative:
What are the two
components of a vector?
What is meant by a
system that is in
equilibrium?
Describe a system that is
in equilibrium / not in
equilibrium.
Summative:
Select
Instructional
Strategies –
Information
(Catch, give and/or
demonstrate
necessary
information,
misconceptions,
etc…)
Catch (5 mins)
Three students each hold a mass (~5 lb weight) at a different
position: straight down, at 45 degree angle, and straight out.
The goal is which student will hold their respective weight the
longest. The student holding the weight straight out should
struggle the most and drop the weight first followed by the
student holding the weight at 45 degree angle.
As the weight moves from straight down to straight out, the
forces and stress on the human body increases exponentially,
thus explaining why the student holding the weight straight
down can hold the weight the longest.
Direct Lesson (20 mins)
The lesson is an application of the force triangle and Law of
Sines. There is a quick review of systems in equilibrium, force
triangles, and Law of Sines. An example used for the review
is to solve is a street light hanging from two poles.
Activity (20 mins)
The activity is described in the Require Learner Participation
section below.
Wrap-Up (5 mins)
Students may question the applicability of this lesson. The
simple problem formulation is seen everywhere in our daily
lives. We use cranes to lift items and this knowledge allows
us to build and use cranes effectively. You will never see a
crane attempt to lift an object while the boom is perfectly
horizontal. The human body is another great example
because our arms are similar to cranes, using muscles as a
pulley system. During the catch, the student holding the
weight horizontal puts a lot of stress on the arm. However,
the student holding the mass vertical had no problem due to
the limited stress on the arm.
Student holds the mass at the
designated angle for as long
as possible.
Misconceptions:
The students may think that it is possible to have the ropes
perfectly horizontal in our system. However, this is impossible
because the graph will illustrate that we have an a symptote.
The tension in the rope goes to infinity as we have an angle
that nears 0 degrees.
Utilize Technology
Calculators
Other Resources
(e.g. Web, books, etc.)
Require
Learner
Participation
Activity
(Describe the
independent activity
to reinforce this
lesson)
Students will draw the free-body-diagram of the system. The
system is a mass being suspended by two ropes from above.
The ropes will form the same angle from the horizon to the
rope. The mass will start low where the angle between the
rope and horizon is near 90 degrees. In small increments, the
mass will rise by changing the length of the rope to the point
that the angle between the rope and horizon is near 0
degrees. At each increment, a student can place a piece of
paper between the weight and board to trace the system onto
the paper. One student will be responsible for one system
setup.
Each student is responsible
for one system setup. The
student will trace the system
(two rope tensions and the
mass) by placing a piece of
paper between the weight
and board. By tracing the
rope, students get a more
accurate angle rather than
trying to free-hand draw the
system.
The students will draw a force triangle based on the system
that they drew. Then they will calculate the tension in each
rope using Law of Sines.
After all stages are completed, the results are graphed by the
relationship of the angle between the vertical and the rope,
and the tension in the rope. The graph will show an
exponential growth shape with an asymptote at 90 degrees.
Thus, proving that it is mathematically impossible to have a
system with the rope straight when a mass is hanging from
the rope.
In order of smallest angle to
largest angle, each student
provides the angle and
tension in the rope to graph
on the board.
This is an example of the graph that is created from the
calculated data. There is an asymptote at 90 degrees which
proves that the ropes can never be perfectly horizontal
whenever there is a mass hanging from the rope.
160
140
Force (N)
120
100
80
60
40
20
0
0
20
40
60
80
Angle
Evaluate
See Appendix A for pre- and post-assessment
(Assessment)
(Steps to check for
student
understanding) – See
Objectives above
Important Attachments:
1. Pre-Post Assessment
2. Worksheets
3. PowerPoint
4. Reflection after lesson
N/A
Additional Notes
Summary of Engineering Fields
Aerospace engineers design, develop, and test aircraft, spacecraft, and missiles and supervise the manufacture of these products. Those
who work with aircraft are called aeronautical engineers, and those working specifically with spacecraft are astronautical engineers. Aerospace
engineers develop new technologies for use in aviation, defense systems, and space exploration, often specializing in areas such as structural
design, guidance, navigation and control, instrumentation and communication, or production methods. They also may specialize in a
particular type of aerospace product, such as commercial aircraft, military fighter jets, helicopters, spacecraft, or missiles and rockets, and
may become experts in aerodynamics, thermodynamics, celestial mechanics, propulsion, acoustics, or guidance and control systems.
Biomedical engineers develop devices and procedures that solve medical and health-related problems by combining their knowledge of
biology and medicine with engineering principles and practices. Many do research, along with life scientists, chemists, and medical scientists,
to develop and evaluate systems and products such as artificial organs, prostheses (artificial devices that replace missing body parts),
instrumentation, medical information systems, and health management and care delivery systems. Biomedical engineers may also design
devices used in various medical procedures, imaging systems such as magnetic resonance imaging (MRI), and devices for automating insulin
injections or controlling body functions. Most engineers in this specialty need a sound background in another engineering specialty, such as
mechanical or electronics engineering, in addition to specialized biomedical training. Some specialties within biomedical engineering include
biomaterials, biomechanics, medical imaging, rehabilitation engineering, and orthopedic engineering.
Chemical engineers apply the principles of chemistry to solve problems involving the production or use of chemicals and biochemicals. They
design equipment and processes for large-scale chemical manufacturing, plan and test methods of manufacturing products and treating
byproducts, and supervise production. Chemical engineers also work in a variety of manufacturing industries other than chemical
manufacturing, such as those producing energy, electronics, food, clothing, and paper. They also work in health care, biotechnology, and
business services. Chemical engineers apply principles of physics, mathematics, and mechanical and electrical engineering, as well as
chemistry. Some may specialize in a particular chemical process, such as oxidation or polymerization. Others specialize in a particular field,
such as nanomaterials, or in the development of specific products. They must be aware of all aspects of chemicals manufacturing and how the
manufacturing process affects the environment and the safety of workers and consumers.
Civil engineers design and supervise the construction of roads, buildings, airports, tunnels, dams, bridges, and water supply and sewage
systems. They must consider many factors in the design process, from the construction costs and expected lifetime of a project to
government regulations and potential environmental hazards such as earthquakes and hurricanes. Civil engineering, considered one of the
oldest engineering disciplines, encompasses many specialties. The major ones are structural, water resources, construction, environmental,
transportation, and geotechnical engineering. Many civil engineers hold supervisory or administrative positions, from supervisor of a
construction site to city engineer. Others may work in design, construction, research, and teaching.
Computer hardware engineers research, design, develop, test, and oversee the manufacture and installation of computer hardware.
Hardware includes computer chips, circuit boards, computer systems, and related equipment such as keyboards, modems, and printers.
(Computer software engineers—often simply called computer engineers—design and develop the software systems that control computers.
These workers are covered elsewhere in the Handbook.) The work of computer hardware engineers is very similar to that of electronics
engineers in that they may design and test circuits and other electronic components, but computer hardware engineers do that work only as it
relates to computers and computer-related equipment. The rapid advances in computer technology are largely a result of the research,
development, and design efforts of these engineers.
Electrical engineers design, develop, test, and supervise the manufacture of electrical equipment. Some of this equipment includes electric
motors; machinery controls, lighting, and wiring in buildings; automobiles; aircraft; radar and navigation systems; and power generation,
control, and transmission devices used by electric utilities. Although the terms electrical and electronics engineering often are used
interchangeably in academia and industry, electrical engineers have traditionally focused on the generation and supply of power, whereas
electronics engineers have worked on applications of electricity to control systems or signal processing. Electrical engineers specialize in areas
such as power systems engineering or electrical equipment manufacturing.
Environmental engineers develop solutions to environmental problems using the principles of biology and chemistry. They are involved in
water and air pollution control, recycling, waste disposal, and public health issues. Environmental engineers conduct hazardous-waste
management studies in which they evaluate the significance of the hazard, advise on treatment and containment, and develop regulations to
prevent mishaps. They design municipal water supply and industrial wastewater treatment systems. They conduct research on the
environmental impact of proposed construction projects, analyze scientific data, and perform quality-control checks. Environmental engineers
are concerned with local and worldwide environmental issues. They study and attempt to minimize the effects of acid rain, global warming,
automobile emissions, and ozone depletion. They may also be involved in the protection of wildlife. Many environmental engineers work as
consultants, helping their clients to comply with regulations, to prevent environmental damage, and to clean up hazardous sites.
Materials engineers are involved in the development, processing, and testing of the materials used to create a range of products, from
computer chips and aircraft wings to golf clubs and snow skis. They work with metals, ceramics, plastics, semiconductors, and composites to
create new materials that meet certain mechanical, electrical, and chemical requirements. They also are involved in selecting materials for
new applications. Materials engineers have developed the ability to create and then study materials at an atomic level, using advanced
processes to replicate the characteristics of materials and their components with computers. Most materials engineers specialize in a
particular material. For example, metallurgical engineers specialize in metals such as steel, and ceramic engineers develop ceramic materials
and the processes for making them into useful products such as glassware or fiber optic communication lines.
Mechanical engineers research, design, develop, manufacture, and test tools, engines, machines, and other mechanical devices. Mechanical
engineering is one of the broadest engineering disciplines. Engineers in this discipline work on power-producing machines such as electric
generators, internal combustion engines, and steam and gas turbines. They also work on power-using machines such as refrigeration and airconditioning equipment, machine tools, material handling systems, elevators and escalators, industrial production equipment, and robots used
in manufacturing. Mechanical engineers also design tools that other engineers need for their work. In addition, mechanical engineers work in
manufacturing or agriculture production, maintenance, or technical sales; many become administrators or managers.
Table 2: Earnings distribution by engineering specialty, May 2006
Lowest
10%
Specialty
Lowest
25%
Median
Highest
25%
Highest
10%
Aerospace engineers
59,610
71,360
87,610
106,450
124,550
Biomedical engineers
44,930
56,420
73,930
93,420
116,330
Chemical engineers
50,060
62,410
78,860
98,100
118,670
Civil engineers
44,810
54,520
68,600
86,260
104,420
Computer hardware
engineers
53,910
69,500
88,470
111,030
135,260
Electrical engineers
49,120
60,640
75,930
94,050
115,240
Environmental engineers
43,180
54,150
69,940
88,480
106,230
Materials engineers
46,120
57,850
73,990
92,210
112,140
Mechanical engineers
45,170
55,420
69,850
87,550
104,900
Table 3: Average starting salary by engineering specialty and degree , 2007
Curriculum
Bachelor's
Aerospace/aeronautical/astronautical
Master's
Ph.D.
$53,408
$62,459
Bioengineering and biomedical
51,356
59,240
$73,814
Chemical
59,361
68,561
73,667
Civil
48,509
48,280
62,275
Computer
56,201
60,000
92,500
Electrical/electronics and communications
55,292
66,309
75,982
Environmental/environmental health
47,960
Materials
56,233
Mechanical
54,128
62,798
72,763
Footnotes:
(NOTE) Source: National Association of Colleges and Employers
Bureau of Labor Statistics, U.S. Department of Labor, Occupational Outlook Handbook, 2008-09 Edition, Engineers, on the Internet
at http://www.bls.gov/oco/ocos027.htm(visited November 20, 2009).
Reflection
This lesson can be done within one day given that the students have a base understanding of force triangles and Law of Sines. It is a great way to
illustrate how these operations are used in everyday applications. If the students struggle with those concepts, I would build in some extra time to
get the students up to speed on the subject matter.
I added one additional catch to the one listed in the lesson plan that helped demonstrate the human body application of the lesson. Prior to the
class starting, I asked a student to carry the weights I was using for the original catch from the back of the class to the front. None of the students,
even the student carrying the weights, caught onto the point of this exercise until I brought it to their attention at the end of the lesson. The human
body muscles are similar to pulleys; our muscles are always pulling and never pushing. The student who brought up the weights carried them at
his side. This position carries the least amount of load (force) on his muscles. So the human body seeks to perform tasks at the most efficient
and less stressed positions, such as carrying an object. In regards to the original catch, try to have the students to be of the same gender. It
works best with the group demonstrating the catch to have roughly the same physically abilities.
Although you can have a system where the tension in the ropes is different by offsetting the location of the weight, it is easiest to keep the weight
in the middle. Therefore, theoretically the tension in both ropes is the same. When the students are tracing the system onto the paper, it becomes
difficult to determine the horizon. So I created a work around to help solve this problem during the lesson. Since the weight is in the middle, have
the students measure the angle between the ropes on the paper with a protractor. Then subtract that number from 180 degrees and then divide
by two. This will give the angle from the horizon to the rope on either side.
Appendix A
Pre- and Post-Assessment
Rope Tension: Rate of Change
Pre- and Post-Assessment
1. Circle all systems that are in equilibrium:
a.
b.
c.
d.
2. Find ‘T’ using Law of Sines
T
35˚
75˚
15N
70˚
25N
3. Explain why the angle α cannot equal 0˚ in the following system
T1
T2
α
α
5N
Rope Tension: Rate of Change
Pre- and Post-Assessment Key
1. Circle all systems that are in equilibrium:
a.
b.
c.
d.
2. Find ‘T’ using Law of Sines
Law of Sines states:
T
sin 75 sin 35 sin 70
=
=
25𝑁
15𝑁
𝑇
35˚
75˚
∴ 𝑇=
15N
70˚
25N
sin 70 ∗ 25𝑁
sin 75
𝑇 = 24.32𝑁
3. Explain why the angle α cannot equal 0˚ in the following system
T1
T2
α
α
5N
The force in T1 and T2 goes to infinity as the
angle α nears 0˚. Therefore, since the system
has an asymptote at 0˚, it is physically
impossible.