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

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!