Developing The Gear Ratio Formula
Developingthe Gear Ratio FormulaBuild KnowledgeINTRODUCTIONWhat Students Do in This ActivityIn this activity students make observations about the rotation relationships in the tables they made during the Recording GearRotations activity. They explore these relationships as constant valuesthat depend on the number of teeth on the two gears. They areintroduced to the concept of ratio as a way to express these relationships. Students derive the formula for finding the gear ratio of a pairof gears: gear ratio teeth in driven gear/teeth in driver gear.RationaleStudents express the gear rotation relationships mathematically. Theyinvestigate the concept of ratio. They find the gear ratio for trains ofsimple gears.Students will engage in the following: record in a table the numbers of rotations of two meshing gears express this rotation relationship as a ratio record in a table the numbers of teeth of two meshing gears express this tooth relationship as a ratio derive the formula for finding the gear ratio from the numbers ofteeth on two meshing gearsTime2–3 class L STUDIESLANGUAGE ARTSMakingConnectionsThis activity should bedone only after theRecording GearRotations activity. It isessential that studentshave experience inmanipulating gears andrecording the rotationrelationships beforelearning ways toexpress these relationships more formally. Ifpossible, talk to theteacher who workedwith the students onthat activity about theirwork.115
Developing the Gear Ratio FormulaMaterials for each student, one copy of the Gear Ratio Recording Table the Gear Rotation Recording Tables students made in theRecording Gear Rotations activityCLASSROOM ACTIVITYActivity DescriptionIn the Recording Gear Rotations activity students observed andrecorded the rotation relationships of each gear in a meshing pair.Ask students to get out the tables they recorded in that activity.Drawing ofGear Pair(driver, driven)1545Teeth onTeeth on Rotations of Rotations ofDriver Gear Driven Gear Driver Gear Driven Gear154531Ask students to consider the 15/45 gear train. Ask: If I rotate the15-tooth driver gear 3 times how many times would the 45-toothdriven gear rotate?Most students will probably now understand that if the 15-tooth gearrotates 3 times the 45-tooth gear will rotate 1 time.Now ask students to give the numbers of rotations for the 45-toothdriven gear for several different numbers of rotations of the 15-toothdriver gear. How many times would the 45-tooth driven gear rotate ifI rotate the 15-tooth driver gear 6 times? 9 times? How do you know?Ask students to explain how they can find the numbers for therotation of the 45-tooth gear given the number of rotations of the15-tooth gear. Help them focus on the relationship between therotating gears as they explain their answers. The relationship of thetwo meshing gears does not change. Their teeth keep them rotatingtogether in the same relative numbers of rotations. The 15-tooth gearalways rotates 3 times for every 1 rotation of the 45-tooth gear.Another way of saying this is that the 15-tooth gear rotates 3 times asfast as the 45-tooth gear. This relationship will always be true nomatter how many times the 15-tooth gear rotates. The relationship ofrotations of the two gears is constant. It does not depend on howmany times the gears rotate, or on how fast they rotate.We use the word ratio to describe this constant relationship betweentwo values.A gear ratio is the ratio of the number of rotations of a driver gear tothe number of rotations of a driven gear.116
Build KnowledgeA colon is often used to show a gear ratio:gear ratio rotations of a driver gear : rotations of a driven gear.SCIENCEFor every rotation of the 45-tooth gear, the 15-tooth gear must rotate 3times. This is true no matter how many times the 45-tooth gearrotates. The ratio between the rotations of the 15-tooth driver gearand the 45-tooth driven gear is 3 to 1. That is, the gear ratio is 3 to 1.Using the colon notation, the gear ratio is 3:1.The order of the two numbers in a ratio is very important. In thiscase, the ratio of the 15-tooth driver gear and the 45-tooth driven gearis 3:1. A gear ratio is always given as the ratio of the rotations of thedriver gear to the rotations of the driven gear. In terms of the bicycleexample the students used earlier, the gear ratio is the ratio of pedalrotations to wheel rotations.MATHEMATICSTECHNOLOGYEDUCATIONSOCIAL STUDIESLANGUAGE ARTSUse another gear pair on the students’ table as an example for furtherexamination: If the 75-tooth driver gear rotates 3 times, how manytimes will a 15-tooth driven gear rotate? What if the 75-tooth drivergear rotates 5 times? How do you know?Again, help the students focus on the relationship between the gears.In this case, the 15-tooth gear rotates 5 times for every rotation of the75-tooth gear. This is true regardless of the number of times the75-tooth gear rotates.What is the gear ratio of a 75-tooth driver gear and a 15-tooth drivengear? The gear ratio is 1:5.Ask students to complete the Teeth Ratio and Gear Ratio columns inthe table.Drawing ofGear Pair(driver, driven)Teeth onDriverGearTeeth onDrivenGearGear RatioTeeth(driverRatio Rotations Rotations rotations:(driver: of Driver of Driven drivendriven)GearGear rotations)Teacher Tip1545154515:45(or 1:3)313:1Facilitating Student ExplorationIt is critical that students see the difference between the ratio of teethand the gear ratio. A gear ratio is the ratio of rotation. A gear withfewer teeth must rotate more times when it meshes with a gear thathas more teeth. If students are unsure of the difference, you can showthem examples using the gears in the materials. For example, the15-tooth gear has to rotate 5 times to turn the 75-tooth gear 1 time.You may want to tellstudents that there areconventions in whichnumbers to use whenexpressing gear ratiosin order to avoidnumbers less than 1. Ifthe driver gear rotatesfaster than the drivengear, the gear ratio isoften given in the form(driver gearrotations):1. If thedriven gear rotatesfaster than the drivergear, the gear ratiois often given in theform 1:(drivengear rotations).117
Developing the Gear Ratio FormulaSharing and InterpretingWhen students have completed the two ratio columns, they candiscuss their findings. Is there a relationship between the gear ratioand the tooth ratio? What is that relationship?Allow students time to express their ideas. Ask them to providereasons for their statements.Help students see how the tooth ratio can be used to find the gearratio. This relationship between the tooth ratio and the gear ratio isvery important. It allows us to find a gear ratio without actuallyhaving to build a gear train and turn the gears. We can find the ratioof any two gears if we know the number of teeth on each of the gears.Show students how the gear ratio can be expressed as a fraction.Write this equation on the board:gear ratio rotations of driver gearrotations of driven gearThis format uses a fraction to express a gear ratio.1545ABFor this gear pair, the gear ratio rotations of gear Arotations of gear B 31This is another way of saying that gear A rotates 3 times while gear Brotates 1 time. It’s the same as saying the gear ratio is 3:1. If we knowthe number of times gear A rotates we can divide that number by 3 tofind the number of times gear B rotates.You can now ask the class: What’s a way to find a gear ratio withoutactually turning the gears?Students may answer that you can use the numbers of teeth on thegears to find the gear ratio.Add the teeth ratio expression to the equation you wrote earlier:gear ratio 118rotations of driver gearrotations of driven gear teeth of driven gearteeth of driver gear
Build KnowledgeHelp students understand that the rotation expression and the teethexpressions are equivalent. You can use the numbers from theabove example in order to make this clearer:gear ratio 31 4515Help students see that they can find the gear ratio using only thenumbers of teeth on the gears. Write the following equation on theboard:gear ratio teeth of driven gearteeth of driver gearSCIENCEMATHEMATICSTECHNOLOGYEDUCATIONSOCIAL STUDIESLANGUAGE ARTSTell students that they will be able to use this formula to find the gearratio for any two gears if they know the number of teeth on bothgears. They can simply divide the number of teeth on the driven gearby the number of teeth on the driver gear to find the gear ratio.In the Using the Gear Ratio Formula activity students will use thisformula to find gear ratios.119
Developing the Gear Ratio FormulaNOTES120
Student Reproducible MasterGEAR RATIO RECORDING TABLEName Design Team DateDrawing ofGear Pair(driver, driven)Teeth onTeeth onDriver Gear Driven en)Gear Ratio(driverrotations:Rotations of Rotations of drivenDriver Gear Driven Gear rotations)121
Student Reproducible MasterGEAR RATIO RECORDING TABLEName Design Team DateDrawing ofGear Pair(driver, driven)Teeth onTeeth onDriver Gear Driven GearTeethRatio(driver:driven)Gear Ratio(driverrotations:Rotations of Rotations of drivenDriver Gear Driven Gear rotations)123
Drawing of Gear Pair (driver, driven) Teeth on Driver Gear Teeth on Driven Gear Rotations of Driver Gear Rotations of Driven Gear 15 45 15 45 3 1 Ask students to consider the 15/45 gear train. Ask: If I rotate the 15-to
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Work out the gear ratios of these gears: (numbers inside the circles are the numbers of teeth on each gear) Blue drive gear, green driven gear, pink idler gear, yellow compound gear 1: Simple meshed gear 2: Simple gear train with idler gear 15 3: Compound gear train 4: Choose the correct words to make these statements true for these gear trains.
Chính Văn.- Còn đức Thế tôn thì tuệ giác cực kỳ trong sạch 8: hiện hành bất nhị 9, đạt đến vô tướng 10, đứng vào chỗ đứng của các đức Thế tôn 11, thể hiện tính bình đẳng của các Ngài, đến chỗ không còn chướng ngại 12, giáo pháp không thể khuynh đảo, tâm thức không bị cản trở, cái được
3. Explain what a gear train does? 4. Explain what a simple gear train is? 5. Explain torque? 6. What does a larger gear driving a smaller gear achieve? 7. What does a smaller gear driving a larger gear achieve? 8. What is the driven gear? 9. What is the driver gear? 10. What is the idler gear? 11. What is the calculation for working out gear .
