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Zorluk: OrtaRatios, Rates, and Proportions

In an industrial mechanical system, Gear AA has 1616 teeth, Gear BB has 2424 teeth, and Gear CC has 4040 teeth. Gear AA is meshed directly with Gear BB, and Gear BB is meshed directly with Gear CC. If Gear AA rotates at a constant speed of 150150 revolutions per minute (rpm\text{rpm}), what is the rotational speed, in revolutions per minute (rpm\text{rpm}), of Gear CC?

Cevap: 60 rpm

Cevap

The rotational speed of Gear C is 60 rpm.
For meshed gears, the linear speed of teeth at the point of contact must be identical. Thus, the product of the number of teeth and rotational speed remains constant (NASA=NCSCN_A S_A = N_C S_C). Substituting the known values gives 16×150=40×SC16 \times 150 = 40 \times S_C, yielding 2,400=40SC2,400 = 40 S_C, so SC=60S_C = 60 rpm.

Adım Adım Çözüm

1
Determine the inverse proportional relationship between number of gear teeth and rotational speed.
The product of teeth count and rotational speed is constant across directly meshed gears: NA×SA=NB×SB=NC×SCN_A \times S_A = N_B \times S_B = N_C \times S_C.
Directly meshed gears engage tooth for tooth, meaning they pass the same total number of teeth per unit time.
2
Calculate the total tooth displacement rate per minute from Gear A.
16×150=2,40016 \times 150 = 2,400 teeth per minute.
Gear A has 16 teeth and completes 150 revolutions per minute.
3
Calculate the rotational speed of Gear C.
SpeedC=2,40040=60\text{Speed}_C = \frac{2,400}{40} = 60 rpm.
Gear C has 40 teeth, so dividing the total tooth displacement rate by 40 yields its revolutions per minute.

Anahtar Kavram

Inverse Proportionality in Gear Rates
Tahmini Süre:1m 15s
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Konular

Ratios, Rates, and Proportions

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