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Difficulty: MediumGears and Gear Ratios

A hand-cranked emergency power generator uses a two-stage compound gear train to increase the rotational speed of its rotor. The hand crank is connected directly to the driving gear, Gear A, which has 6060 teeth. Gear A meshes with Gear B, which has 1515 teeth. Gear B shares a solid intermediate shaft with Gear C, which has 4040 teeth. Finally, Gear C meshes with Gear D on the generator's rotor shaft, which has 1010 teeth.

If a user turns the hand crank at a steady rate of 30 rpm30\text{ rpm}, what is the resulting rotational speed of the generator rotor in rpm\text{rpm}?

Answer: 480 rpm

Answer

480
The correct answer is 480 because the total speed multiplication of a compound gear train is the product of the individual gear ratios of each stage. Stage 1 has a ratio of 60:15 (a 4:1 multiplier). Stage 2 has a ratio of 40:10 (a 4:1 multiplier). The total multiplier is 4 * 4 = 16. The input speed of 30 rpm multiplied by 16 gives a final output speed of 480 rpm.

Step-by-Step Solution

1
Calculate the speed multiplication factor of the first gear pair.
Factor of 4
Gear A (60 teeth) drives Gear B (15 teeth). The ratio of teeth is 60/15 = 4. Because a larger gear is driving a smaller gear, the rotational speed increases by this factor.
2
Calculate the rotational speed of the intermediate shaft.
120 rpm
Multiply the input speed (30 rpm) by the first stage factor (4). 30 * 4 = 120 rpm. Both Gear B and Gear C spin at this exact rate since they are mounted on the same solid shaft.
3
Calculate the speed multiplication factor of the second gear pair.
Factor of 4
Gear C (40 teeth) drives Gear D (10 teeth). The ratio is 40/10 = 4. Again, a larger gear is driving a smaller one, increasing the speed.
4
Calculate the final rotational speed of the rotor.
480 rpm
Multiply the intermediate shaft speed (120 rpm) by the second stage factor (4). 120 * 4 = 480 rpm.

Key Concept

Calculating output speed in a compound gear train
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Topics

Gears and Gear Ratios

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