For the gear train shown below, if the input torque at gear d is 10 Nm, what is the output torque at gear a. Na=61, Nb=49, Nc=20, Nd=46

To find the output torque at gear \( a \) based on the input torque at gear \( d \), we can use the relationship between the gears in the train. The torque transformation can be determined using the gear ratios.

1. **Calculate Gear Ratios**:
   The gear ratio between two gears is given by the formula:
   \[
   \text{Gear Ratio} = \frac{N_{\text{input}}}{N_{\text{output}}}
   \]
   where \( N \) represents the number of teeth on each gear.

2. **Define Gears**:
   - \( a \) (output gear)
   - \( b \)
   - \( c \)
   - \( d \) (input gear)

   Given:
   - \( N_a = 61 \)
   - \( N_b = 49 \)
   - \( N_c = 20 \)
   - \( N_d = 46 \)

3. **Torque Calculation**:
   The relationship for the torques based on gear ratios is:
   \[
   T_a = T_d \cdot \left( \frac{N_d}{N_c} \right) \cdot \left( \frac{N_c}{N_b} \right) \cdot \left( \frac{N_b}{N_a} \right)
   \]
   This simplifies to:
   \[
   T_a = T_d \cdot \frac{N_d}{N_a}
   \]
   since the intermediate gears \( b \) and \( c \) cancel out.

4. **Plugging in Values**:
   Given \( T_d = 10 \, \text{Nm} \):
   \[
   T_a = 10 \, \text{Nm} \cdot \frac{46}{61}
   \]

5. **Calculating Output Torque**:
   \[
   T_a = 10 \, \text{Nm} \cdot 0.7541 \approx 7.54 \, \text{Nm}
   \]

Thus, the output torque at gear \( a \) is approximately **7.54 Nm**.

### Questions to Consider:
1. How do changes in gear sizes affect the output torque?
2. What would happen if gear \( d \) had a different torque?
3. How would the output torque change if gear \( a \) had more teeth?
4. Can you explain how gear trains are used in mechanical systems?
5. What factors influence the efficiency of a gear train?

### Tip:
Always check for the number of teeth on each gear to ensure accurate calculations, as small changes can significantly impact torque and speed ratios.

Discover how to calculate the output torque at gear a given an input torque of 10 Nm at gear d in a gear train with specific teeth counts. This problem illustrates the application of gear ratios and torque transformation, making it ideal for high school students studying physics and mechanics.

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