Mechanical Aptitude
This module is all about the value of a few simple machines,
and how they relate to firefighting. The way to master this
material is to make the connections between the machines
and the advantage they give a firefighter on the job. You can
do that by learning a few formulas that govern the machines.
Think of it this way, no firefighter is going to stand at the
scene of a fire trying to calculate how much force will be
needed to move a beam—that’s not realistic. The firefighter
does need to recognize that it will be quicker and easier to
move that beam using a lever than it will using bare hands.
The formulas are primarily to show the testing agency that
you understand the principles of simple machines.
Mechanical Aptitude
Simple
Machines
Machines are devices comprised of inter-related parts with separate and
distinct functions that are used to make something move (e.g., motors,
sewing machines).
Simple machines change the direction of a force without using energy
(See Fig 1). They include:
•
•
•
•
•
•
•
Levers
Wheels
Pulleys
Screws
Wedges
Inclined Planes
Gears/Belt Drives
Lever
Pulley
Wedge
Screw
Inclined Plane
Wheel
Fig. 1
Levers
Levers:
•
Have 3 parts:
o Fulcrum
o Weight or Resistance
o Effort or Force
•
Fall into 3 classes:
o 1st Class – Comprised of a fulcrum positioned somewhere
between the weight and the resistance (weight) and the
effort (force) (see Fig 2). The effort and distance from the
fulcrum can be varied (e.g., claw hammer or crowbar).
Fig. 2
Resistance or
Effort or
Force
Weight
Fulcrum
MA-2
Mechanical Aptitude
Levers
•
The effort of the 1st class lever always changes.
•
The closer the weight is to the fulcrum, the
less effort it will take to move it.
o 2nd Class – The fulcrum is at one end with the effort (force)
at the other. The resistance (weight) is distributed
somewhere between the two (see Fig 3). Changing the
distribution of the weight will increase or decrease the effort
needed to lift the load. Examples include a wheelbarrow and
a human doing push-ups.
•
A 2nd class lever does not change the direction of the effort
Fig. 3
Resistance or
Weight
Effort or
Force
Fulcrum
F
i
o 3rd Class – The effort is applied between the fulcrum and the
g
weight (see Fig 4).
.
Fig. 4
Effort or
Force
3
Resistance or
Weight
Fulcrum
MA-3
Mechanical Aptitude
Levers Con’t.
A 3rd class lever does not change the direction of the effort.
The human arm is an example of a 3rd class lever—the elbow is the
fulcrum, the hand holds the weight, and the bicep expends the effort to
move the load.
•
3rd class levers are not for heavy jobs they are used to gain speed
•
Other 3rd class levers include shovels and brooms.
Formulas:
When using simple machines, it may be necessary to determine the
mechanical advantage (MA). That just means the weight of the load in
relation to the effort needed to move it (ratio). The MA formulas for the
different classes of levers are shown below:
1St Class Lever
Fig. 5
Load
Effort
MA =
90 lbs
30 lbs
Fulcrum
F
The table below can be used to determine any unknown component
i
(X) in this formula
g
.
If x is:
Then:
MA
Divide the resistance by the amount of
5
effort needed
Divide the weight of the load by the
MA
Effort
Resistance or weight of
the load
Multiply the effort needed by the MA
MA-4
Mechanical Aptitude
Levers Con’t.
Practice Problem 1:
What is the mechanical advantage (MA) for the data shown
in Fig. 5 (p. MA - 4)?
MA =
Resistance
=
Effort
90
30
=3
MA = 3
2nd Class Lever:
Force x Effort Distance = Weight x Resistance Distance
Practice Problem 2:
In the example below (see Fig. 6), the effort distance, the
resistance distance, and the weight of the load are given.
Solve for X to determine the force needed to lift the load.
Fig. 6
Effort Distance = 4′
Effort
Load = 100 lbs
Resistance Distance (to the
center of the load) = 2′
Fulcrum
If x is:
First:
Then:
Effort needed
Multiply the weight of the
load by the resistance
distance
Divide the result by
the effort distance
Use the formula MA =
MA-5
Load
Effort
to determine MA
Mechanical Aptitude
Now answer the following questions using the figures in Fig. 6.
Levers Con’t.
C
A. How much effort is needed to move the load?
100 lbs X 2
o
n
4′
B. What is the MA?
’
100 lb
t
50 lb
.
= 50 lbs of effort needed
MA = 2
3rd Class Lever:
Effort (Force) Distance
= MA
Resistance Force (load) Distance
In the example in Fig. 7, the Effort distance divided by the resistance
distance is a simple reduction in terms.
Fig. 7
Effort Distance
2″₺ 20″₺ Resistance Distance
Effort distance
Wheels and
Axles
=
2″
Resistance Distance =
20″
MA =
1
10
Wheels and Axles:
•
A wheel is a circular frame or disk that revolves on an axis
(commonly found on vehicles and machines).
MA-6
Mechanical Aptitude
Wheels and
Axles Con’t.
•
An axle is the smaller wheel or shaft inside a larger wheel. The
movement of the wheel around the axle moves the object
Used in concert, wheels and axles make moving a load easier and more
efficient. As the larger wheel turns, it engages the smaller wheel (axle)
and together, they move the load with a minimum of effort (see Fig 8).
When the larger wheel is turned it causes rotation in the smaller wheel or
torque. The amount of torque is proportionate to the amount of force
placed on the larger wheel (see Fig 8).
Fig. 8
Wheel radius = 5″
Axle radius = 1″
5″
The formula for finding the MA for a wheel and axle is:
MA=
Wheel Radius
Axle Radius
5″
In Fig. 8, MA =
or 5
1″
•
The larger the wheel, the more force it can apply to the axle.
Examples of wheels and axles can be found on steering wheels,
wheelchairs, door knobs, and faucets handles.
Formulas for circles and other shapes can be found in the Math Formula
Table (See Appendix (p APX – 7)
MA-7
Mechanical Aptitude
A pulley is simply a lever in disguise. A pulley is a wheel and axle with a
Pulleys
channel around the circumference. A rope, belt, chain, or cable is fed into
the channel placing the effort on one side of the pulley, and the resistance
or weight on the other (see Fig. 9).
Fig. 9
Channel
Weight
Effort
Single Pulley
There are 3 kinds of pulleys:
•
•
•
Single (Fixed)
Moveable
Compound (A system of fixed and moveable pulleys)
The single pulley (see Fig. 9) is attached to a single, fixed location such
as a wall or beam. Because the rope or cable is just fed around the wheel,
you have a 1 to 1 result when you expend effort (i.e., if you pull the rope
down 2 feet, your load will move up 2 feet; if the rope is pulled down 7
feet, the load will be lifted 7 feet.)
•
A single pulley changes the direction of the effort.
•
There is no mechanical advantage or MA for a
single pulley
The moveable pulley does what the name implies—it moves with the
effort expended. The effort necessary to move a load with any pulley
is dictated by the number of ropes deployed around the wheels.
MA-8
Mechanical Aptitude
A single pulley uses one wheel, and therefore employs a single rope. The
Pulleys Con’t.
moveable pulley engages 2 or more wheels—increasing the amount of
C
rope that displaces the weight of the load. In other words, the resistance
oforce caused by the weight of the load is more evenly distributed over the
nlength of the rope making the load easier to lift (see Fig. 10).
’Fig. 10
t
Effort
Resistance
Force
.
Moveable Pulley
Weight
A moveable pulley does not change the direction of the effort.
The compound pulley is a system of single and moveable pulleys
working together to change the direction of the effort, and decrease the
effort needed in order to move the weight attached.
Fig. 11
In the most common compound pulley—a
block and tackle—a number of single pulleys
are attached to a single axle and then housed in a
‘block’ which is then attached to a fixed location
(wall, ceiling, etc.) A similar assembly is then attached
to a moveable pulley with rope or cable
(see Fig. 11).
A single worker using this configuration can move very heavy loads.
MA-9
Mechanical Aptitude
The compound pulley changes the direction of the effort,
Pulleys Con’t.
C o
and decreases the effort needed to lift the load
How can you easily determine
the effort needed to lift a load?
20 kg Fig.
n12
effort m Input force
’
distance
t
4 m .
A. Simply divide the weight
of the load by the
number of ropes in the
pulley system (do not
count the rope that goes to
the effort).
Example: in Fig. 12, the person
lifting the 200 kg load has to exert
2 m a pull equal to only 50kg
200 kg (200kg divided by 4 ropes).
O
u
Practice Problem 3:
t
Using the four pulley system in Fig. 12, what is the mechanical
p
advantage?
u
Load
200 kg
t
MA
=
or
so
Effort
50 kg
MA = 4
f
o find the efficiency of a pulley system, you must first find mechanical
To
r
advantage)
and the velocity ratio (see note). Then divide the MA by the
c and multiply the result by 100% to arrive at the efficiency rating.
VR
e
Note: Velocity ratio is the ratio of the distance moved by the effort applied,
to the distance moved where the load or resistance is applied. Velocity
d
ratio is sometimes called distance ratio.
i
s
t
a
n
c
e
MA-10
Mechanical Aptitude
Pulleys Con’t.
Practice Problem 4:
C
MA =
o
Use the data in Fig. 12 (previous page) for the following formula:
Resistance force (load)
=
Effort (force)
200 kg
n
’ Velocity Ratio =
t
.
Efficiency =
Effort (force) distance
MA
Load distance
=
=4
50 kg
4
=
4
2
=2
Efficiency = 2
2
VR
=
A screw is a simple machine with an incline that wraps around the shaft
Screws
(threads). The distance between the threads equals the pitch of the screw
(see Fig. 13). The pitch determines the distance the screw will advance
with each rotation.
To find the mechanical advantage of a screw use:
MA =
F
Circumference of the screwdriver handle
i
Pitch of the screw
g
O
Effort (force) distance
MAR=
.
Resistance force (load) distance
Pitch = resistance force
Screwdriver handle circumference = effort distance
3
Practice problem 5:
If you have a screw with 8 threads per inch, and a screwdriver
handle with a radius of 1″ what is the MA?
Hint: Use the table on p. MA-12 to solve the problem
MA-11
1
Mechanical Aptitude
Screws Con’t.
C
Steps:
Formula:
o Find the
Example:
C=2r
2 x 3.14 x 1 = 6.28
circumference of the
n screwdriver handle
’
t
Determine the pitch
of the screw
1 ″ ÷ Threads per inch
1″ ÷ 8 = 0.125
Find the MA
circumference / pitch
6.28 ÷ 0.125 = 50.24
.
Note: Pi (Α ) is always 3.14
Inclined Plane
The inclined plane is a plane surface set at an angle on a horizontal
surface. The angle of the plane is not a right angle. This simple
machine lets the user overcome a large resistance by applying a
relatively small force over a distance longer than the load is to be
raised.
To find the mechanical advantage of the inclined plane use the following
formula:
MA = Length of slope
Height of slope
OR
Effort (force) distance
MA =
O Resistance force (load) distance
R
Practice Problem 6:
Using data from Fig. 14, compute the MA for this inclined plane.
MA =
MA =
Length of slope
Height of slope
15′
3′
4ʹ′ MA = 5
15ʹ′ 200 lbs
Fig. 14
3′
MA-12
Mechanical Aptitude
Inclined Plane
Con’t.
C
Note: The height of a slope will never be greater than the length
of the inclined plane. That means that the MA will never be less
than 1.
o
n
’
Practice Problem 7:
Using the data in Fig 15, determine the effort required to move a
box to the top of the incline.
t
200 lbs
5′
.
20′
Using the formula:
Effort x Effort Distance = Resistance Force x Resistance Distance
F
i
g
.
OR
Force x Length of Inclined Plane = Load x Height
Solve for X
1
5
X (effort) x 20 = 5 x 200
1000
Effort =
So Effort = 50 pounds
20
Wedges
Wedges are very similar to inclined planes with two exceptions:
1. Wedges move while inclined planes are stationary.
2. The effort is applied parallel to the slope on an inclined plane, while
the effort is applied at a right angle to the direction of travel with a
wedge.
There are two types of wedges:
•
Single wedges – are inclined planes with a sharpened edge.
They are commonly used for door stops or wheel chocks
MA-13
Mechanical Aptitude
•
Wedges
Con’t.
Double wedges - have two faces that meet in a sharply acute
angle. They are generally used to split objects by applying a
C
downward force that pushes the object outward in opposing
o
directions (e.g., an ax or a chisel).
n
Can you think of a common object that is an exception to the normal force
’ distribution of a wedge?
t
A. A nail is a double wedge that actually joins objects together
.
rather than splitting them apart.
Wedges change the direction of the force. They are the simplest
machines with a mechanical advantage to increase force.
The mechanical advantage of wedges can be increased just by
sharpening the tapered edge.
To determine the MA of a single wedge, use the formula for an inclined
plane:
Length of slope
MA =
Height of slope
For a double wedge, use the formula:
MA =
Effort (force) distance
Resistance force (load) distance
Practice Problem 8:
Using the data in Fig. 16, what is the MA for the double
wedge shown?
MA =
4 cm
18 cm (effort distance)
4 cm (resistance force distance
MA = 4.5 cm
18 cm
MA-14
Mechanical Aptitude
Gears and
Belt Drives
A wedge with a longer more tapered point
F requires less
effort when cutting through an object (e.g.,
i splitting wood is
easier if the ax used has a long, slender ghead).
. to generate
Gears are wheels with teeth or cogs that mesh together
torque and motion. Their advantage over a pulley is that they don’t slip.
1
Gears either turn a shaft, or they are turned by a shaft.6
Gears may:
•
•
•
Improve force
Limit force
Change the direction of the force
Spur gears (see Fig 17) are the most common gears. They are designed
with the teeth mounted on a parallel shaft or axle.
Fig. 17
The configuration of these gears causes each gear to rotate in an opposite
direction (see Fig 18). In this figure, one gear acts as the driver, the other
is driven.
A
B
Fig. 18
MA-15
Mechanical Aptitude
Gears and
Belt Drives
Con’t.
•
C
o
When the smaller gear is the driver, the torque is increased
while the rotation is slowed. (or more power less speed)
•
When the larger gear is the driver, rotation speed is
increased, but torque declines. (Less power more speed)
nWhen figuring gear ratio, one rotation of the larger gear is always the
’ basis for the comparison—regardless of which
t
one is the driver. That means it is always expressed as 1
. The gear ratio is determine by using the formula:
Fig. 19
Distance moved by the effort
Driven = 60
teeth
Distance moved by the load
OR in this example:
Input (Driver Gear)
30 teeth
Output (Driven Gear)
60 teeth
Driver = 30
teeth
The ratio is expressed as 2:1 meaning the small gear has to turn two
times for every revolution of the large gear. If the large gear were the
driver, it would be 1:2
Think of a ratio as a comparison rather than a division problem—that is,
how many times the small wheel has to rotate in proportion to the large
one. The result is always expressed with the driver number first.
MA = # of teeth on the driver: # of teeth on the driven
MA-16
Mechanical Aptitude
Practice Problem 9:
Gears and
Belt Drives
Con’t.
1. What is the MA for the gears in Fig. 19
MA =
30 teeth
or 2:1
20 teeth
2. What is the MA if the large gear is the driver?
MA =
20 teeth
or 1:2
30 teeth
In a gear train—3 or more gears working in concert—the driver sets the
rotation sequence for the rest of the gears.
In Fig 20, gear A is the driver and it is rotating clockwise. Gears B and C
both touch A so both rotate counter clockwise. Gear D touches B and C so
it rotates clockwise.
A
B
C
D
Pedal and sprocket gears are most commonly found in bicycles. They
have longer teeth than a typical gear, and interact with a chain. When the
large gear at the front (pedal) rotates, it turns the chain around the smaller
(sprocket) gear in the rear to propel the bicycle.
MA-17
Pedal and sprocket gears rotate in the same direction
Use the formula for gears for the pedal and sprocket MA
Mechanical Aptitude
Belt Drives are pulley wheels connected by flat flexible bands. Like the
Gears and
Belt Drives
Con’t.
pedal and sprocket, the belt drive wheels rotate in the same direction.
Conveyor belts are large industrial belt drives used to move materials from
Cone place to another.
o There are two kinds of belt drive
n
Drive:
’Positive
Characteristics:
Teeth in the belt interlock with teeth in the wheel to
t
limit slipping.
.Non-positive
Smooth belts are configured with a “v” shape to
provide a larger contact area with the wheel.
Use the formula for gears for the Belt Drive MA.
When the driver is smaller than the driven wheel, there will be a
mechanical advantage.
Use the bubble sheet on pg. MA-25 to answer the questions on the
following pages.
MA-18
Mechanical Aptitude
Review
Questions
1. The figure at the right is an example of:
a. Winch
b. Pulley
c. Inclined plane
d. Platform
2. A compound pulley has:
a. A single wheel
b. Chains instead of ropes
c. A system of fixed and moveable pulleys working together
d. Longer ropes than a single pulley
3. The seesaw is an example of:
a. A first class lever
b. A compound lever
c. A third class lever
d. A wedge
4. The mechanical advantage of a single pulley is:
a. 0
b. 1
c. 2
d. 3
5. What is the mechanical advantage for the figure
shown here?
a. 1
b. 2
c. 3
d. 4
MA-19
100 lb 25
lbs
Mechanical Aptitude
Review
Questions
6. Your arm is an example of a:
a. 1st class lever
b. 2nd class lever
c. 3rd class lever
d. A fulcrum
7. To find the mechanical advantage of a wheel and axle, use the
formula Wheel Radius divided by:
a. Axle radius
b. Axle circumference
c. Amount of torque
d. Revolutions per minute
8. A compound pulley:
a. Increases effort but does not change the direction
b. Changes the direction of the effort and decreases the
effort
c. Changes the direction of the effort and increases the effort
d. Increases the effort and changes the direction of the effort
9. A block and tackle is an example of:
a. A compound pulley
b. A single pulley
c. Belt drive
d. Drive train
10. If you’re trying to turn a screw with pitch of 10 using a screwdriver
with a handle radius of 1 inch, what is the mechanical advantage?
a. 75
b. 0.1
c. 6.28
d. 62.8
MA-20
Mechanical Aptitude
Review
Questions
11. What is the mechanical advantage
3ʹ′ for the figure shown at the right?
12ʹ′ a. 3
b. 50
c. 4
9 ʹ′ 100 lbs
d. 5
12. Which of the following statements about a wedge is true?
a. A wedge has no mechanical advantage
b. A wedge does not change the direction of the force
c. The mechanical advantage of a wedge can be increased
by sharpening the tapered edge
d. Wedges are only made with a single face
13. When the teeth in the belt interlock with teeth in the
wheel to limit slipping this is referred to as what kind
of belt drive?
a. Class 1
b. Class 2
c. Positive
d. Non-positive
14. Pedal and sprocket gears are most common in:
a. Cars
b. Bicycles
c. Wishing wells
d. Skates
MA-21
Mechanical Aptitude
Review
Questions
15. Gears can do all of the following except:
a. Improve force
b. Limit force
c. Change the direction of the force
d. Eliminate the force
16. Since the height of a slope will never be greater than the length of
the inclined plane, the mechanical advantage:
a. Will always be less than 1
b. Will be a whole number
c. Will be an even number
d. Will never be less than 1
17. In the figure at the right, if the load weighs
400 kg what is the efficiency of the
pulley?
10 m
a. 1
b. 2
c. 3
5m
d. 4
18. In the figure at the right, what
is the gear ratio if A is the driver?
a. 1:3
b. 1:2
c. 3:1
d. 2:1
MA-22
Mechanical Aptitude
Review
Questions
19. Which one of the following pairs is not representative of a double
wedge?
a. Block and tackle
b. Nail and chisel
c. Axe and slotted screwdriver
d. Knife and scissors
20. The pivot point of a lever is called the:
a. Force
b. Resistance
c. Fulcrum
d. Apex
MA-23
Mechanical Aptitude
Answer Key
1. C
2. C
3. A
4. A
5. D – 100 lbs/25 lbs = an MA of 4
6. C
7. A
8. D
9. A
10. D
11. C - 12' (length of the incline) / 3' (height of platform) = MA of 4
12. C
13. C
14. B
15. D
16. D
17. B
18. B – A (the large gear) is the driver and has 24 teeth, divide that by
the 12 teeth in B (the small gear). The small gear has to turn twice
for every single revolution of the large gear (24/12 = 2). Since the
large gear is the driver, its revolution is expressed first in the ratio
19. A
20. C
MA-24
Mechanical Aptitude
Answer Sheet
MA-25
Mechanical Aptitude
Answer Sheet