1. Chapter 2: Mechanical Systems
Engineering Mathematics

2. Introduction/Description
The mechanical systems lesson will cover the simple machines and the mechanical advantage of the lever simple machine.
Student teams will complete a mechanical advantage team project.

3. Chapter 2: Outline Introduction to Simple Machines
Calculating Weight and Friction
Mechanical Advantage and Friction of a Simple Machine
Calculating Power and Efficiency in Complex Machines

4. Objectives and Result
Students will understand the meaning of mechanical systems.
Students will identify the different types of mechanical systems and the mechanical advantage of mechanical systems and simple machines.
Students will understand the careers and educational opportunities available in the mechanical engineering industry.
The result will be that student complete a mechanical advantage team project.

5. Schedule of Assignments
Class/Period(s)
Topic
Reading
Assignment
1-2
Introduction
Vocabulary
Mechanical
Engineering
O*Net
Chapter 2.1
#1-Individual Write a one-page paper about the Mechanical Engineering Occupation.
3-10
Simple Machines
Chapter 2.2
#2-In teams of 2-3, complete the Simple Machines Challenge.
11-15
Levers and Pulleys
Chapter 2.3
#3-In teams of 2-3, complete the Lever and Pulley Challenge.
16-20
Mathematical
Principles Challenge
Chapter 2.4
#4-In teams of 2-3, complete the Mathematical Principles Challenge.

6. Vocabulary Mechanical Systems Mathematical Principles Simple Machines
Lever
Wheel and Axle
Pulley
Inclined Plane
Wedge
Screw
Mechanical Systems – machines that use energy to perform some activity
Mathematical Principles – principles taught and learned in an equitable manner in a setting that ensures that problem solving, reasoning, connections, communication, and conceptual understanding are all developed simultaneously along with procedural fluency
Simple Machines – mechanical systems that change the direction or magnitude of a force; the term refers to the six classical simple machines, which were defined by Renaissance scientists
Lever – a simple machine consisting of a rigid bar that rests on a fulcrum, or pivot
Wheel and Axle – a simple machine consisting of a wheel connected to an axle so that turning the wheel also turns the axle
Pulley – a simple machine consisting of a grooved wheel that is turned by a rope or chain
Inclined Plane – a simple machine consisting of a uniform sloped surface
Wedge – simple machine consisting of an angled object used to separate two objects, lift an object, or hold an object in place
Screw – simple machine consisting of an inclined plane wrapped around an axis

7. Vocabulary, cont. Linkages Cams Turnbuckles Gears Key Fasteners
V-belt Drives
Chain Drives
Fulcrum
Moment Arm
Force Arm
Resistance Arm
Linkages – include garage door mechanisms, car wiper mechanisms, gear shift mechanisms
Cams – rotating or sliding pieces in a mechanical linkage used especially in transforming rotary motion into linear motion or vice-versa
Turnbuckles – devices for adjusting the tension or length of ropes, cables, tie rods, and other tensioning systems
Gears – rotating machine parts having cut teeth, or cogs, which mesh with another toothed part in order to transmit torque. Two or more gears working in tandem are called a gear train or a gear drive.
Key Fasteners – any of various devices, as a snap or hook and eye, for holding together two objects or parts sometimes required to be separate, as two edges or flaps of a piece of clothing
V-belt Drives – belts with a flat bottom and tapered sides, used to transmit motion between two pulleys
Chain Drives – two or more gears connected with a chain that provide a way of transmitting mechanical power from one place to another
Fulcrum – the center or axis of rotation of the system
Moment Arm – the distance from any force or weight that produces torque to the fulcrum
Force Arm – the distance from an applied force to the fulcrum (the moment arm of the force)
Resistance Arm – the distance from the resistance to the fulcrum (the moment arm of the resistance)

8. Mechanical Engineering
Mechanical engineering is a discipline of engineering that applies the principles of physics and materials science for analysis, design, manufacturing, and maintenance of mechanical systems.
It is the branch of engineering that involves the production and usage of heat and mechanical power for the design, production, and operation of machines and tools.
It is one of the oldest and broadest engineering disciplines.

9. Career/Educational Opportunities
Career and educational opportunities include the following:
Mechanical engineers
Mechanical engineering technologists
Architectural and engineering managers
Electromechanical engineering technologists
Mechatronics technologists
The careers and educational opportunities choices include the following:
Mechanical engineers – Bachelor’s Degree in Mechanical Engineering
Mechanical engineering technologists – Bachelor’s Degree in Mechanical Engineering, Mechanical Technology, or Mechanical Technician
Architectural and engineering managers
Bachelor’s Degree Computer Science — Computer Engineering, General; Computer Hardware Engineering; Computer Software Engineering
Bachelor’s Degree Engineering — Aerospace, Aeronautical and Astronautical Engineering; Agricultural/Biological Engineering and Bioengineering; Architectural Engineering; Architecture; Biomedical/Medical Engineering; Ceramic Sciences and Engineering
Electromechanical engineering technologists – Bachelor’s Degree Engineering in Electromechanical Technology/Electromechanical Engineering Technology
Mechatronics engineers
Bachelor’s Degree Computer Science — Computer Hardware Engineering
Bachelor’s Degree Engineering — Computer Hardware Engineering

10. Mechanical Engineering Assignment
Visit the O*Net website (
Write a one-page essay on the mechanical engineering profession.
Discuss at least one sub-specialty of mechanical engineering in your essay.
Use the O*Net website and at least one other primary source.

11. Simple Machines
Simple machines are mechanical systems that change the direction or magnitude of a force. The term refers to the six classical simple machines, which were defined by Renaissance scientists.
Lever – a simple machine consisting of a rigid bar that rests on a fulcrum, or pivot
Wheel and Axle – a simple machine consisting of a wheel connected to an axle so that turning the wheel also turns the axle

12. Simple Machines, cont.
Pulley – a simple machine consisting of a grooved wheel that is turned by a rope or chain
Inclined Plane – a simple machine consisting of a uniform sloped surface
Wedge – a simple machine consisting of an angled object used to separate two objects, lift an object, or hold an object in place
Screw – a simple machine consisting of an inclined plane wrapped around an axis
Pulley – a simple machine consisting of a grooved wheel that is turned by a rope or chain
Inclined Plane – a simple machine consisting of a uniform sloped surface
Wedge – a simple machine consisting of an angled object used to separate two objects, lift an object, or hold an object in place
Screw – a simple machine consisting of an inclined plane wrapped around an axis

13. Simple Machines Challenge
In teams of 2-3 students,
complete and sign the team contract spreadsheet for this unit,
build a simple machine,
compare and contrast the simple machine built by your team with a simple machine built by another team in your class, and
calculate the mechanical advantage of each type of simple machine built by each team in the class.

14. What are Mechanical Systems?
Mechanical systems are machines that use energy to perform some activity.
Mechanical systems manage power to accomplish a task that involves forces and movement.
Mechanical systems are devices having parts that perform or assist in performing any type of work.

15. Types of Mechanical Systems
The most commonly used mechanical systems are listed below.
Levers
Linkages
Cams
Turnbuckles
Pulleys
Gears
Key Fasteners
V-belt Drives
Chain Drives

16. Levers
Levers are rigid bars that exert a force to move a load by turning on a pivot or fulcrum.
Levers are classified systems of torque; the moment of a force or system of forces tending to cause rotation around a fixed point.

17. Levers, cont.
Relative positions of force, resistance, and axis of rotation vary in the different types or classes of levers.
As with any torque calculations, operations on levers determine the tendency for some force to produce rotation around a fixed point.

18. Components of a Lever
Fulcrum: the center or axis of rotation of the system
Moment Arm: the distance from any force or weight that produces torque to the fulcrum
Force Arm: the distance from an applied force to the fulcrum (the moment arm of the force)
Resistance Arm: the distance from the resistance to the fulcrum (the moment arm of the resistance)

19. Classes of Levers
In a first class lever, the applied force and the resistance are on opposite sides of the fulcrum.
In a second class lever, the resistance is between the applied force and the fulcrum.
In a third class lever, the applied force is between the resistance and the fulcrum.

20. First Class Lever applied force resistance resistance arm force arm
fulcrum
applied force
resistance arm
resistance
force arm
In a first class lever, the applied force and the resistance are on opposite sides of the fulcrum.

21. Second Class Lever force arm resistance arm resistance applied force
fulcrum
resistance
applied force
resistance arm
force arm
In a second class lever, the resistance is between the applied force and the fulcrum.

22. Third Class Lever resistance arm force arm resistance applied force
fulcrum
resistance
applied force
resistance arm
force arm
In a third class lever, the applied force is between the resistance and the fulcrum.

23. Torque Produced in Lever Systems
Two torques are produced in lever systems:
By the applied force
By the resistance
The direction in which a lever system moves is dependent on the relative lengths of the force and resistance arms, as well as the magnitudes of force and resistance.

24. Mechanical Advantage (MA) of Levers
The effectiveness of a lever at moving a resistance is a calculated value:
Mechanical Advantage
𝐹𝑜𝑟𝑐𝑒 𝑂𝑢𝑡 𝐹𝑜𝑟𝑐𝑒 𝐼𝑛 or =
Mechanical Advantage video
Mechanical Advantage

25. Mechanical Advantage of Levers, cont.
Because of their different configurations, the mechanical advantage of
a first class lever can favor the force or resistance depending on the placement of the fulcrum,
a second class lever always favors the force arm, and
a third class lever always favors the resistance arm.

26. MA: First Class Lever
fulcrum
applied force
resistance arm
resistance
force arm
The fulcrum in a first class lever system can often vary in position to favor the force arm or the resistance arm.

27. MA: Second Class Lever
fulcrum
resistance
applied force
resistance arm
force arm
In a second class lever system, the mechanical advantage favors the force arm. The force arm will always be longer.

28. MA: Third Class Lever
fulcrum
resistance
applied force
resistance arm
force arm
The mechanical advantage of a third class lever system favors the resistance arm. The resistance arm is always longer.

29. Lever and Pulley Challenge
In teams of 2-3 students,
research lever and pulley systems,
design a lever or pulley system to do some type of useful work,
describe the type or class of lever or pulley system you selected, and
present your lever or pulley system to the class and explain why you chose that type of system to do the work you selected.

30. Linkages
Linkages include garage door mechanisms, car wiper mechanisms, gear shift mechanisms.
They are an important part of mechanical engineering.

31. Cams
Cams are rotating or sliding pieces in a mechanical linkage used especially in transforming rotary motion into linear motion or vice-versa.
A common example is the camshaft of an automobile, which takes the rotary motion of the engine and translates it into the reciprocating motion necessary to operate the intake and exhaust valves of the cylinders.

32. Turnbuckles
Turnbuckles are devices for adjusting the tension or length of ropes, cables, tie rods, and other tensioning systems.

33. Pulleys
Pulleys are wheels on an axle that are designed to support movement of a cable or belt along its circumference.
Pulleys are used in a variety of ways to lift loads, apply forces, and to transmit power.

34. Gears
Gears are rotating machine parts having cut teeth, or cogs, which mesh with another toothed part in order to transmit torque.
Two or more gears working in tandem are called a gear train or a gear drive.

35. Key Fasteners
Key fasteners are any of various devices, as a snap or hook and eye, for holding together two objects or parts sometimes required to be separate, as two edges or flaps of a piece of clothing.

36. V-belt Drives
V-belt drives are belts with a flat bottom and tapered sides, used to transmit motion between two pulleys.
Multiple V-belts are often used together in order to increase carrying power.

37. Chain Drives
Chain drives, or sprocket drives, provide a way of transmitting mechanical power from one place to another.
They are often used to convey power to the wheels of a vehicle, particularly bicycles and motorcycles.
They are also used in a wide variety of machines besides vehicles.

38. Mathematical Principles Challenge
In teams of 2-3 students, calculate the mechanical advantage for the mechanical systems listed below:
Levers
Linkages
Cams
Turnbuckles
Pulley systems
Gear drives
Key fasteners
V-belt drives
Chain drives
Teacher Tip:
Presentations should be less than 10 minutes in length and you should try to schedule all teams to present on the same day in order to avoid plagiarism (most calculations will be similar due to the nature of the assignment).
Present your calculations to the class.
Write a five-page paper documenting your findings.

39. Credits ClipArt; http://www.clipart.com/en/
Images;
Slide 24
Mechanical Advantage video; from YouTube user; Science Online;