A compound reverted gear train (shown in the image) is to be used. We want to make sure the input and output shafts are aligned. If Gear 2 = 50 teeth, Gear 3 = 15 teeth, and Gear 4 =40 teeth, how many teeth should Gear 5 have? X
The Correct Answer and Explanation is :
To solve this, we need to understand the working of the compound reverted gear train. A reverted gear train has gears that mesh with each other, and the shafts are aligned. In this scenario, gears 2, 3, and 4 are fixed on one shaft, while gear 5 is attached to another shaft.
The basic principle to solve this problem involves the gear ratio. The input and output gears must have a relationship in terms of their number of teeth to ensure the shafts align properly. Gear 2 meshes with Gear 3, and Gear 4 meshes with Gear 5. Since Gear 2, Gear 3, and Gear 4 are on the same shaft, their rotational speeds are the same.
Step-by-Step Calculation:
- Let the number of teeth on Gear 5 be denoted as ( T_5 ).
- From the problem, we know the following:
- Gear 2 has 50 teeth.
- Gear 3 has 15 teeth.
- Gear 4 has 40 teeth.
The formula for gear ratio is:
[
\text{Gear ratio} = \frac{\text{Number of teeth on driven gear}}{\text{Number of teeth on driving gear}}
]
For the gear train, the rotational speeds of gears on the same shaft are equal. This means that the ratio of the teeth between Gear 2 and Gear 3 should be the same as the ratio of teeth between Gear 4 and Gear 5.
Thus:
[
\frac{T_2}{T_3} = \frac{T_4}{T_5}
]
Substituting the known values:
[
\frac{50}{15} = \frac{40}{T_5}
]
Now, solving for ( T_5 ):
[
T_5 = \frac{40 \times 15}{50} = 12
]
Answer:
Gear 5 should have 12 teeth.
Explanation:
In a compound reverted gear train, the teeth of the gears on the same shaft must maintain a proportional relationship to ensure alignment of the shafts. By equating the ratios of teeth between the relevant gears, we determined that Gear 5 should have 12 teeth to maintain the correct alignment and functionality of the gear system. This ensures that the input and output shafts are correctly aligned, allowing the gear train to function as intended.
Now, I’ll generate the image for you.
Here is the image of the compound reverted gear train with the gears and their teeth numbers as discussed. It should help illustrate how the gears mesh and interact with each other. Let me know if you need further clarification or have any other questions!
