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Difficulty: Very hardInverse Proportions

A military transport vehicle's winch system uses a gear train consisting of three gears. The driving gear (Gear A) has 6060 teeth and rotates at 150 RPM150\text{ RPM}. It meshes with an idler gear (Gear B) which has 4545 teeth. Gear B meshes directly with the driven gear (Gear C). If Gear C must rotate at a speed of 5 revolutions per second5\text{ revolutions per second} to safely operate the winch, how many teeth must Gear C have?

  1. 30Answer
  2. B
    120
  3. C
    300
  4. D
    1800

Answer

30 teeth
The number of teeth on a gear and its rotational speed are inversely proportional. In a gear train, an idler gear (Gear B) only changes the direction of rotation, not the mathematical ratio between the first and last gears (TA×SA=TC×SCT_A \times S_A = T_C \times S_C). First, ensure the speed units match by converting Gear C's speed to RPM: 5 rev/s×60 seconds/minute=300 RPM5\text{ rev/s} \times 60\text{ seconds/minute} = 300\text{ RPM}. Then, equate the products for Gear A and Gear C: 60×150=TC×30060 \times 150 = T_C \times 300. Solving for TCT_C gives 9000÷300=309000 \div 300 = 30 teeth.

Step-by-Step Solution

1
Identify the relationship in a gear train.
The product of teeth and rotational speed is constant across meshed gears (T1×S1=T2×S2T_1 \times S_1 = T_2 \times S_2). The idler gear (Gear B) does not change the final relationship between Gear A and Gear C.
To establish the mathematical formula needed and eliminate unnecessary variables.
2
Convert Gear C's speed to the same unit as Gear A (RPM).
5 revolutions/second×60 seconds/minute=300 RPM5\text{ revolutions/second} \times 60\text{ seconds/minute} = 300\text{ RPM}.
Both gears must use the exact same rate units before an accurate proportion can be calculated.
3
Set up the inverse proportion equation connecting Gear A to Gear C.
60 teeth×150 RPM=TC×300 RPM60\text{ teeth} \times 150\text{ RPM} = T_C \times 300\text{ RPM}.
To isolate the unknown variable based on the constant product rule of gears.
4
Solve the equation for the number of teeth on Gear C (TCT_C).
9000=300×TC9000 = 300 \times T_C, which simplifies to TC=9000÷300=30 teethT_C = 9000 \div 300 = 30\text{ teeth}.
To find the final required dimension for the driven gear.

Key Concept

Inverse Proportions in Compound Mechanics

Alternative Method

You can solve this in two steps by finding the idler gear's speed first. Gear B's speed: 60×150=45×SB→9000=45×SB→SB=200 RPM60 \times 150 = 45 \times S_B \rightarrow 9000 = 45 \times S_B \rightarrow S_B = 200\text{ RPM}. Then relate Gear B to Gear C (converted to 300 RPM300\text{ RPM}): 45×200=TC×300→9000=300×TC→TC=3045 \times 200 = T_C \times 300 \rightarrow 9000 = 300 \times T_C \rightarrow T_C = 30.
Estimated Time:1m 45s
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Topics

Inverse Proportions

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