# Wheels Radius / Distance Traveled

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1 Mechanics Teacher Note to the teacher On these pages, students will learn about the relationships between wheel radius, diameter, circumference, revolutions and distance. Students will use formulas relating these measurements to compute the distance a wheel of given radius travels for a given number of wheel revolutions. Students will have to use both fractions and decimals to make these calculations. While the worksheet is designed to help students learn the geometry of the circle and the relationship between wheel size, revolutions and distance, and may be completed by students with little background in these areas, the existing ability to multiply fractions and decimals 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 there are no instructions regarding rounding. The answers assume rounding to digits beyond the decimal place, except for known fractions. Teachers may wish to supply additional instructions. If they do not, students answers will vary slightly according to what rounding conventions they use. The illustration below explains and reinforces the relationships between radius, diameter and circumference. is the measurement of a straight line from the center of a circle to the edge. will always equal one half diameter. is the measurement of a straight line across the center of an object (in this case a wheel). The circumference of a circle is the total distance around its outside. equals the diameter of the circle times, which is about 3.4. One revolution of a wheel will make it move a distance equal to its circumference. 5.4

2 Teacher Mechanics An example is shown below. Complete the information on the following pages. Example If the wheel below makes one revolution, how far will it go? Use the tables below to find the answer /6" After reviewing the concepts, students are expected to use the following procedure: Identify the radius written below a pictured wheel (.) Enter it in the box labeled Multiply the radius by to calculate the diameter and enter this number in the box labeled Multiply this diameter by π, simplified to 3.4, to find the circumference Find the number of revolutions specified in the instructions (.) Enter it in the box labeled Multiply this number by the number calculated for circumference to find the distance the wheel will travel Approximate classroom time: 0-0 minutes depending on students background 5.5

3 Mechanics Teacher Students successfully completing the worksheet will be able to:. Identify radius from a labeled drawing. Describe the geometry of a circle 3. Describe the relationship between radius, diameter, circumference, revolutions and distance for a wheel 4. Calculate diameter from radius 5. Calculate circumference from diameter 6. Calculate distance from wheel circumference and revolutions 7. Multiply decimals and fractions Standards addressed: Math Standards Numbers and Operations Algebra Geometry Measurement Problem Solving Technology Standards The Nature of Technology Standards ; Design 8, 9 Note: Workbook answers begin on the next page. 5.6

4 Teacher Mechanics is the measurement of a straight line from the center of a circle to the edge. will always equal one half diameter. is the measurement of a straight line across the center of an object (in this case a wheel). The circumference of a circle is the total distance around its outside. equals the diameter of the circle times, which is about 3.4. One revolution of a wheel will make it move a distance equal to its circumference. An example is shown below. Complete the information on the following pages. Example If the wheel below makes one revolution, how far will it go? Use the tables below to find the answer..565" x = 3.5" 3.5" x 3.4 = 9.8" 9 /6" 9.8" x = 3 9.8" 5.7

5 Mechanics Teacher If the wheel below completes.75 revolutions, how far will it go? Use the tables below to find the answer /3" If the wheel below completes 3 /6 revolutions, how far will it go? Use the tables below to find the answer /8"

6 Teacher Mechanics If the wheel below completes.75 revolutions, how far will it go? Normally it s much easier to calculate decimal numbers, rather than fractions, so the first thing we ll do is convert the fraction to a decimal. 5 /3" converts to.47". We know that the circumference the distance a wheel will travel in one revolution is equal to π x D where D is the diameter. Since the, R, is equal to half of the diameter, we can change the equation from C = π x D to: C = π x (R), or, x π x R. 5 /3" If we substitute the values in the equation, we can see that: C = x 3.4 x.47" =.95". Since the wheel makes.75 revolutions, the total distance traveled is: distance =.95 inches / revolution x.75 revolutions = 5.6 inches. If the wheel below completes 3 /6 revolutions, how far will it go? As we saw earlier, calculations are much easier when using decimals, rather than fractions. So the first thing we ll do is convert both numbers, the radius and the number of revolutions, to decimals. The radius of 5 /8" converts to 0.65 and the number of revolutions converts from 3 /6 to.9. 5 /8" To find the circumference, we ll use the equation from the previous problem, C = x π x R or C = x 3.4 x.65" = 3.93". Since the wheel makes.9 revolutions, the total distance traveled is: distance = 3.93 inches / revolution x.9 revolutions = 8.6 inches. Again, because of round off, your students answers may be slightly different. 5.9

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