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To find the speed of gear \( N_a \) in rpm given that gear \( N_d \) is rotating at 100 rpm, we can use the gear ratio relationships in the gear train.

The formula for gear speed is:
\[
\text{Speed ratio} = \frac{N_{\text{driven}}}{N_{\text{driver}}}
\]
where \( N \) is the number of teeth on the gear.

### Step-by-Step Calculation:

1. **Given Data:**
   - \( N_a = 94 \) teeth
   - \( N_b = 66 \) teeth
   - \( N_c = 33 \) teeth
   - \( N_d = 18 \) teeth
   - Speed of \( N_d \) (output gear) = 100 rpm

2. **Calculate Gear Speed Ratios:**
   - Since gear \( N_d \) drives gear \( N_c \), their speed ratio is given by:
     \[
     \frac{\text{Speed of } N_c}{\text{Speed of } N_d} = \frac{N_d}{N_c} = \frac{18}{33}
     \]
     Hence, the speed of \( N_c \) is:
     \[
     \text{Speed of } N_c = 100 \times \frac{33}{18} = 183.33 \text{ rpm}
     \]

3. **Calculate the Speed of \( N_b \):**
   - Gear \( N_c \) drives gear \( N_b \), and their speed ratio is:
     \[
     \frac{\text{Speed of } N_b}{\text{Speed of } N_c} = \frac{N_c}{N_b} = \frac{33}{66}
     \]
     Hence, the speed of \( N_b \) is:
     \[
     \text{Speed of } N_b = 183.33 \times \frac{66}{33} = 91.67 \text{ rpm}
     \]

4. **Calculate the Speed of \( N_a \):**
   - Finally, gear \( N_b \) drives gear \( N_a \), and their speed ratio is:
     \[
     \frac{\text{Speed of } N_a}{\text{Speed of } N_b} = \frac{N_b}{N_a} = \frac{66}{94}
     \]
     Hence, the speed of \( N_a \) is:
     \[
     \text{Speed of } N_a = 91.67 \times \frac{94}{66} \approx 130.56 \text{ rpm}
     \]

Therefore, the speed of gear \( N_a \) is approximately 130.56 rpm.

Would you like further details or have any questions?

### Related Questions:
1. How would the speed change if the number of teeth on \( N_a \) increased?
2. What is the effect of reversing the driving and driven gears on the gear ratio?
3. How can we calculate the torque for each gear in this gear train?
4. What are the potential applications of gear trains in machinery?
5. How would backlash affect the performance of this gear train?

### Tip:
In gear trains, smaller gears rotate faster than larger gears when meshed together, following the inverse relationship between the number of teeth and rotational speed.

For the gear train shown in the image, if the Number of teeth are Na=94, Nb=66, Nc=33, and Nd=18, what is the speed of Na in rpm if Nd is rotating at 100 rpm?

Learn how to calculate the speed of gear Na in a gear train when Nd rotates at 100 rpm. This problem uses gear ratios and rotational speed proportionality, with teeth counts Na=94, Nb=66, Nc=33, and Nd=18. Suitable for Grades 11-12.

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