Project ForgeTheory of Machines
Governors & Flywheels
GATE 2015
Hard
A pinion with radius r1 and inertia I1 is driving a gear with radius r2 and inertia I2. Torque τ1 is applied on the pinion. θ1, θ2 are the angular displacements of pinion and gear respectively, with contact forces F1 (on pinion) and F2 (on gear). Which of the following relations hold true?
- AF1≠F2; τ1=I1θ1̈; F2=I2(r1/r2²)θ1̈
- F1=F2; τ1=[I1+I2(r1/r2)²]θ1̈; F2=I2(r1/r2²)θ1̈
- CF1=F2; τ1=I1θ1̈; F2=I2(1/r2)θ2̈
- DF1≠F2; τ1=[I1+I2(r1/r2)²]θ1̈; F2=I2(1/r2)θ2̈
Explanation
By Newton's third law, the tooth contact force is the same magnitude on both gears: F1=F2=F. Rolling contact gives r1θ1=r2θ2, so θ2̈=(r1/r2)θ1̈. For the gear: F·r2=I2θ2̈=I2(r1/r2)θ1̈, so F=I2(r1/r2²)θ1̈. For the pinion: τ1=I1θ1̈+F·r1=I1θ1̈+I2(r1²/r2²)θ1̈=[I1+I2(r1/r2)²]θ1̈.
Source: GATE ME 2015 (Session 1) (2015)
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