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To solve this problem, we’ll apply the principles of gear ratio in a gear train. We are given that the input torque at gear \( d \) is 10 Nm, and we need to find the output torque at gear \( a \).

Here's the approach:

### Step 1: Gear Ratio Calculation
For gears in a train, the torque ratio is inversely proportional to the gear ratio. The gear ratio between two meshed gears \( i \) and \( j \) is given by:
\[
\text{Gear Ratio} = \frac{N_j}{N_i}
\]
where \( N \) denotes the number of teeth on each gear.

Given dataTo solve this problem, we’ll apply the principles of gear ratio in a gear train. We are given that the input torque at gear \( d \) is 10 Nm, and we need to find the output torque at gear \( a \).

Here's the approach:

### Step 1: Gear Ratio Calculation
For gears in a train, the torque ratio is inversely proportional to the gear ratio. The gear ratio between two meshed gears \( i \) and \( j \) is given by:
\[
\text{Gear Ratio} = \frac{N_j}{N_i}
\]
where \( N \) denotes the number of teeth on each gear.

Given 
- \( N_a = 61 \)
- \( N_b = 49 \)
- \( N_c = 20 \)
- \( N_d = 46 \)
- Torque at \( d \): \( T_d = 10 \, \text{Nm} \)

### Step 2: Calculate Total Gear Ratio from Gear \( d \) to Gear \( a \)
The gear train follows a sequence:
- \( d \) to \( c \),
- \( c \) to \( b \),
- \( b \) to \( a \).

#### Intermediate Gear Ratios
1. **From \( d \) to \( c \)**:
   \[
   \text{Gear Ratio}_{d \to c} = \frac{N_d}{N_c} = \frac{46}{20} = 2.3
   \]

2. **From \( c \) to \( b \)**:
   \[
   \text{Gear Ratio}_{c \to b} = \frac{N_b}{N_c} = \frac{49}{20} = 2.45
   \]

3. **From \( b \) to \( a \)**:
   \[
   \text{Gear Ratio}_{b \to a} = \frac{N_a}{N_b} = \frac{61}{49} \approx 1.2449
   \]

#### Total Gear Ratio from \( d \) to \( a \)
Since the gears are in series, the total gear ratio from \( d \) to \( a \) is the product of each intermediate gear ratio:
\[
\text{Total Gear Ratio} = 2.3 \times 2.45 \times 1.2449 \approx 7.020
\]

### Step 3: Calculate Output Torque at Gear \( a \)
The output torque at gear \( a \) can be found by multiplying the input torque at gear \( d \) by the total gear ratio:
\[
T_a = T_d \times \text{Total Gear Ratio}
\]
\[
T_a = 10 \, \text{Nm} \times 7.020 = 70.2 \, \text{Nm}
\]

### Answer
The output torque at gear \( a \) is approximately:
\[
\boxed{70.2 \, \text{Nm}}
\]

Would you like more details on any part of this solution, or have further questions?

### Related Questions:
1. How would the output torque change if the input torque at gear \( d \) was increased?
2. What would happen to the gear ratio if we swapped gears \( b \) and \( c \)?
3. How does the number of teeth on each gear affect the overall gear train torque?
4. Could you use the same method to find the output speed at gear \( a \) if the input speed at gear \( d \) was given?
5. How would adding another gear between \( c \) and \( b \) affect the torque at \( a \)?

#### Tip:
In gear trains, adding more gears in the sequence can significantly change the output torque and speed, depending on the number of teeth in each gear.

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

Learn how to calculate the output torque in a gear train system with an input torque of 10 Nm applied at gear d, with gears a, b, c, and d having tooth counts Na=61, Nb=49, Nc=20, and Nd=46 respectively. This solution explores the concept of gear ratios and torque multiplication in mechanical systems, providing a step-by-step guide.

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