Teacher Mechanics
Gears Ratios and Speed / Problem Solving
Note to the teacher
On this page, students will learn about the relationship between gear ratio, gear rotational speed, wheel radius, diameter,
circumference, revolutions and distance. Students will determine gear ratio by dividing the number of teeth on the driven
gear by the number of teeth on the driving gear. They will determine the rotational speed of the driving gear by multiplying the gear ratio by the rotational speed of the driven axle. They will determine distance by multiplying this rotational
speed by the wheel circumference. They will also have to know how to determine circumference from diameter and
radius. Students will have to use both fractions and decimals to make these calculations. While the worksheet is designed
to help students learn how to determine gear ratio and derive rotational speed from it, and to derive distance from wheel
circumference and rotational speed, and may be successfully completed by students with little background in these areas,
the existing ability to multiply and simplify fractions, the ability to use decimals, and the ability to determine circumference
from diameter or radius will be necessary to successfully complete the worksheet. Teachers may wish to review any or all
of these skills depending on their students’ background.
Note that this exercise is more challenging than the exercises in Gears 1-4. To successfully complete the worksheet, students
must have a working understanding of most of the concepts and equations introduced in both the Gears and Wheels worksheets. Students will also have to perform more calculations than in any of the previous Gears and Wheels worksheets.
Note that there are no instructions regarding rounding. The answers assume rounding to 2 digits beyond the decimal
place, except for known fractions. Teachers may wish to supply additional instructions. If they do not, students’ answers
may vary slightly according to what rounding conventions they use.
The formulas are:
(# Teeth on driven gear)
1
(# Teeth on driving gear)
Wheel
Circumference
=
Gear Ratio
3
9.82”
Revolutions
x
1
(Speed of driving gear)
2
(Gear Ratio) X (Speed of driven gear) = (Speed of driving gear)
4
(Gear Ratio)
Distance
=
9.82”
= (Speed of driven gear)
The necessary formulas (above) are included with the instructions to reinforce the concepts.
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©2005 Robomatter Inc. RE 2.5_RW 1.1
Mechanics Teacher
Gears Ratios and Speed / Problem Solving
A.
B.
C.
1. Assuming the circumference of the wheel and the RPM of the motor are exactly the same for all experiments,
which gear ratio would create the most speed, A, B or C? Why did you choose that answer?
The gear ratio “B” is lowest, meaning it would create the highest speed for the driven axle.
2. If the driving gear is moving at 100 RPM, how fast will the driven gear move for pictures A, B and C?
A. 58.3 RPM B. 300 RPM C. 100 RPM
Students are expected to use the following procedure:
•
•
•
•
•
•
•
•
Identify gear sizes by counting teeth
Determine the gear ratio of each pair of gears
Identify the information required by the question
Reconstruct the equations provided as an example
Enter the data provided into these equations
Manipulate the equations if necessary
Solve the equations
Note that the first question tests comprehension of the concept of gear ratios.
Some students may have more difficulty with this question than the others, which only
require entering numbers in equations and solving them.
Approximate classroom time: 15-25 minutes depending on students’ background
©2005 Robomatter Inc. RE 2.5_RW 1.1
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Teacher Mechanics
Gears Ratios and Speed / Problem Solving
Students successfully completing the worksheet will be able to:
1. Define gear ratio
2. Understand the concept of gear ratio, in particular that low gear ratios will create more wheel speed than
high gear ratios when motor speed is held constant
3. Identify gear size by counting teeth
4. Simplify fractions
5. Describe the geometry of a circle
6. Describe the relationship between radius, diameter, circumference, revolutions and distance for a wheel
7. Calculate diameter from radius
8. Calculate circumference from diameter
9. Derive driving gear RPM from driven gear RPM if given the gear ratio
10. Calculate distance from wheel circumference and revolutions
11. Multiply decimals and fractions
12. Reconstruct equations relating diameter, circumference, revolutions and distance
13. Identify data provided in word problems
14. Identify information required in the word problems
15. Manipulate these equations to solve for the required information
Standards addressed:
Math Standards
Numbers and Operations
Algebra
Geometry
Measurement
Problem Solving
Connections
Technology Standards
The Nature of Technology Standard 1
Design Standards 8, 9
Abilities for a Technological World Standard 12
Using Technology to Design the Future Standards 16, 18, 19
Science Standards
Content Standard B
Content Standard E
Note: Workbook answers begin on the next page.
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©2005 Robomatter Inc. RE 2.5_RW 1.1
Mechanics Teacher
Gears Ratios and Speed / Problem Solving
Instructions
Use the formulas and pictures below to answer the following questions
Note: The driving gear is always on the right. Possible gear sizes are 40, 24, 14 and 8 tooth gears.
The formulas are:
(# Teeth on driven gear)
1
(# Teeth on driving gear)
Wheel
Circumference
=
Gear Ratio
3
9.82”
Revolutions
x
1
(Speed of driving gear)
2
(Gear Ratio) X (Speed of driven gear) = (Speed of driving gear)
4
(Gear Ratio)
A.
B.
=
1. Assuming the circumference of the wheel and the RPM of the motor are exactly the same for all experiments,
which gear ratio would create the most speed, A, B or C? Why did you choose that answer?
We know that the lower the gear ratio, the faster the driven gear will revolve for every revolution of the driving
gear. By counting teeth, we know that the lowest gear ratio is the one in picture B.
To find the answer to this question, we can use Formula 4.
For picture A, we can see that the gear ratio is 24/14, which simplifies to 12/7,
and we know from the question that the driving gear is moving at 100 RPM.
So—100 RPM/(12/7) = 100 RPM x 7/12 = 58.3 RPM.
For picture B, we can see that the gear ratio is 8/24, which simplifies to 1/3,
and we know from the question that the driving gear is moving at 100 RPM.
So—100 RPM/(1/3) = 100 RPM x 3 = 300 RPM.
For picture C, we can see that the gear ratio is 24/24, which simplifies to 1/1,
and we know from the question that the driving gear is moving at 100 RPM.
So—100 RPM/(1) = 100 RPM.
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9.82”
= (Speed of driven gear)
C.
2. If the driving gear is moving at 100 RPM, how fast will the driven gear move for pictures A, B and C?
Distance
Teacher Mechanics
Gears Ratios and Speed / Problem Solving
3. If the driving gear moves at 100 RPM and the circumference of the wheel is 4.4 cm, how far will the robot move in
2 minutes for pictures A, B and C?
To find the answer to this question, we can use Formula 4 and then Formula 3.
For picture A, we can see that the gear ratio is 24/14, which simplifies to 12/7, and we know from the question
that the driving gear will move 100 revolutions/minute x 2 minutes = 200 revolutions. So 200 revolutions/(12/7) =
200 revolutions x 7/12 = 116.6 revolutions. Now we can use formula 3, above. So—116.67 x 4.4 cm = 513.35 cm.
For picture B, we can see that the gear ratio is 8/24, which simplifies to 1/3, and we know from the question
that the driving gear will move 100 revolutions/minute x 2 minutes = 200 revolutions. So 200 revolutions/(1/3)
= 200 revolutions x 3 = 600 revolutions. Now we can use formula 3, above. So—600 x 4.4 cm = 2640 cm.
For picture C, we can see that the gear ratio is 1/1, and we know from the question that the driving gear will
move 100 revolutions/minute x 2 minutes = 200 revolutions. So 200 revolutions/1 = 200 revolutions
Now we can use formula 3, above. So—200 x 4.4 cm = 880 cm.
4. If the driving gear moves at 200 RPM and the diameter of the wheel is 1.9 cm, how far will the robot move in
3 minutes for pictures A, B and C?
To find the answer to this question, we can use Formula 4 and then Formula 3.
For picture A, we can see that the gear ratio is 24/14, which simplifies to 12/7, and we know from the question
that the driving gear will move 200 revolutions/minute x 3 minutes = 600 revolutions. So 600 revolutions/(12/7)
= 600 revolutions x 7/12 = 350 revolutions. The wheel circumference is π x diameter, or π x 1.9 = 5.97 cm.
Now we can use formula 3, above. So—350 x 5.97 cm = 2089.5 cm.
For picture B, we can see that the gear ratio is 8/24, which simplifies to 1/3, and we know from the question
that the driving gear will move 200 revolutions/minute x 3 minutes = 600 revolutions. So 600 revolutions/(1/3)
= 600 revolutions x 3 = 1800 revolutions. The wheel circumference is π x diameter, or π x 1.9 = 5.97 cm.
Now we can use formula 3, above. So—1800 x 5.97 cm = 10746 cm.
For picture C, we can see that the gear ratio is 1/1 and we know from the question that the driving gear will
move 200 revolutions/minute x 3 minutes = 600 revolutions. So 600 revolutions/(1) = 600 revolutions.
The wheel circumference is π x diameter, or π x 1.9 = 5.97 cm. Now we can use formula 3, above.
So—600 x 5.97 cm = 3582 cm.
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©2005 Robomatter Inc. RE 2.5_RW 1.1
Mechanics Teacher
Gears Ratios and Speed / Problem Solving
5. If the driving gear moves at 200 RPM and the radius of the wheel is 2.5 cm, how far will the robot move in
3 minutes for pictures A, B and C?
To find the answer to this question, we can use Formula 4 and then Formula 3.
For picture A, we can see that the gear ratio is 24/14, which simplifies to 12/7, and we know from the question
that the driving gear will move 200 revolutions/minute x 3 minutes = 600 revolutions. So 600 revolutions/(12/7)
= 600 revolutions x 7/12 = 350 revolutions. The wheel circumference is 2 x π x r = 2 x π x 2.5 cm = 15.7 cm.
Now we can use formula 3, above. So—350 x 15.7 cm = 5495 cm.
For picture B, we can see that the gear ratio is 8/24, which simplifies to 1/3, and we know from the question
that the driving gear will move 200 revolutions/minute x 3 minutes = 600 revolutions. So 600 revolutions/(1/3)
= 600 revolutions x 3 = 1800 revolutions. The wheel circumference is 2 x π x r = 2 x π x 2.5 cm = 15.7 cm.
Now we can use formula 3, above. So—1800 x 15.7 cm = 28260 cm.
For picture C, we can see that the gear ratio is 1/1, and we know from the question that the driving gear will
move 200 revolutions/minute x 3 minutes = 600 revolutions. So 600 revolutions/(1) = 600 revolutions. The
wheel circumference is 2 x π x r = 2 x π x 2.5 cm = 15.7 cm.
Now we can use formula 3, above. So—600 x 15.7 cm = 9420 cm.
©2005 Robomatter Inc. RE 2.5_RW 1.1
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