A hand-cranked emergency siren uses a compound gear train to amplify the rotational speed … — Armed Services Vocational Aptitude Battery (ASVAB)

A hand-cranked emergency siren uses a compound gear train to amplify the rotational speed of the crank. The crank is directly attached to Gear A, which has $50$ teeth. Gear A meshes with Gear B, which has $10$ teeth. Gear B shares a solid shaft with Gear C, which has $40$ teeth. Finally, Gear C meshes with Gear D, which has $10$ teeth and is directly attached to the siren's rotor. If the operator turns the crank at a constant speed of $30\text{ rpm}$ , what is the rotational speed of the siren's rotor?

Answer: 600 rpm

Answer

600 rpm

The system consists of two gear pairs that both increase speed. The first pair (

50

teeth driving

10

teeth) increases the speed by a factor of

5

. The second pair (

40

teeth driving

10

teeth) increases the speed by a factor of

4

. The total speed multiplier is

5 \times 4 = 20

. Multiplying the input speed of

30\text{ rpm}

20

yields the final output speed of

600\text{ rpm}

Step-by-Step Solution

Determine the speed multiplier for the first gear pair.

Ratio =

50 / 10 = 5

Gear A drives Gear B. A larger gear driving a smaller gear increases rotational speed proportionally to their tooth count.

Determine the speed multiplier for the second gear pair.

Ratio =

40 / 10 = 4

Gear C drives Gear D. Both pairs are independent in tooth count, but Gear C rotates at the same speed as Gear B because they share a shaft.

Calculate the total mechanical advantage for speed.

Total multiplier =

5 \times 4 = 20

In a compound gear train, the total speed ratio is the product of the individual gear pair ratios.

Calculate the final output speed.

30 \times 20 = 600\text{ rpm}

Multiply the input speed by the total speed multiplier to find the output speed.

Key Concept

Compound Gear Trains and Speed Ratios

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