3.2 Aear with 20 teeth is attached to a 6 KW motor and rotates at 500 rpm . Th gear ratio between the drive gear and the driven gear is $$ 1: 4 $$. If no loss take place, calculate the following: 3.2.1 The number of teeth of the driven gear 3.2.2 The speed of the driven gear 3.2.3 The torque developed by the driver

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To solve the problem, we will break it down into the three parts as specified.

### Known Conditions:
- Number of teeth on the driver gear ($$T_d$$) = 20 teeth
- Power of the motor ($$P$$) = 6 kW
- Speed of the driver gear ($$N_d$$) = 500 rpm
- Gear ratio between the drive gear and the driven gear = $$1:4$$

### 3.2.1 The number of teeth of the driven gear
The gear ratio is defined as the ratio of the number of teeth on the driven gear ($$T_r$$) to the number of teeth on the driver gear ($$T_d$$). Given the gear ratio of $$1:4$$, we can express this as:

$$
\frac{T_d}{T_r} = \frac{1}{4}
$$

From this, we can rearrange to find $$T_r$$:

$$
T_r = 4 \cdot T_d
$$

Substituting the known value of $$T_d$$:

$$
T_r = 4 \cdot 20
$$

Now, let's calculate $$T_r$$.

### 3.2.2 The speed of the driven gear
The speed of the driven gear ($$N_r$$) can be calculated using the gear ratio. The relationship between the speeds of the driver and driven gears is given by:

$$
\frac{N_d}{N_r} = \frac{T_r}{T_d}
$$

Rearranging gives:

$$
N_r = N_d \cdot \frac{T_d}{T_r}
$$

Substituting the known values:

$$
N_r = 500 \cdot \frac{20}{T_r}
$$

### 3.2.3 The torque developed by the driver
The torque ($$T_d$$) developed by the driver can be calculated using the formula:

$$
P = \frac{2 \pi N_d T_d}{60}
$$

Rearranging gives:

$$
T_d = \frac{P \cdot 60}{2 \pi N_d}
$$

Now, let's perform the calculations for $$T_r$$, $$N_r$$, and $$T_d$$.

I'll calculate these values now.
Calculate the value by following steps:
- step0: Calculate:
  $$4\times 20$$
- step1: Multiply the numbers:
  $$80$$
Calculate or simplify the expression $$ (6 * 1000) * 60 / (2 * \pi * 500) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(6\times 1000\right)\times 60}{\left(2\pi \times 500\right)}$$
- step1: Remove the parentheses:
  $$\frac{6\times 1000\times 60}{2\pi \times 500}$$
- step2: Multiply the terms:
  $$\frac{360000}{2\pi \times 500}$$
- step3: Multiply the terms:
  $$\frac{360000}{1000\pi }$$
- step4: Reduce the fraction:
  $$\frac{360}{\pi }$$
Calculate or simplify the expression $$ (500 * (20 / 80)) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$500\left(\frac{20}{80}\right)$$
- step1: Reduce the fraction:
  $$500\times \frac{1}{4}$$
- step2: Reduce the numbers:
  $$125\times 1$$
- step3: Simplify:
  $$125$$
### Calculated Results:

1. **Number of teeth of the driven gear ($$T_r$$)**:
   $$
   T_r = 80 \text{ teeth}
   $$

2. **Speed of the driven gear ($$N_r$$)**:
   $$
   N_r = 125 \text{ rpm}
   $$

3. **Torque developed by the driver ($$T_d$$)**:
   $$
   T_d = \frac{360}{\pi} \text{ Nm} \approx 114.59 \text{ Nm}
   $$

### Summary of Results:
- **Number of teeth on the driven gear**: 80 teeth
- **Speed of the driven gear**: 125 rpm
- **Torque developed by the driver**: Approximately 114.59 Nm

If you need further assistance or additional calculations, feel free to ask!
- **Driven gear teeth**: 80 - **Driven gear speed**: 125 rpm - **Driver torque**: ≈114.59 Nm

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