To calculate the power lost due to friction in the bearing, we can follow these steps:
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Calculate the radius of the shaft: \[ \text{Diameter} = 250 \text{ mm} = 0.250 \text{ m} \] \[ \text{Radius} (r) = \frac{\text{Diameter}}{2} = \frac{0.250 \text{ m}}{2} = 0.125 \text{ m} \]
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Calculate the friction force: The friction force (\( F_f \)) can be calculated using the formula: \[ F_f = \mu \cdot F_n \] where:
- \( \mu \) = coefficient of friction = 0.02
- \( F_n \) = normal force (or axial load on the bearing) = 75 kN = 75000 N
Substituting the values: \[ F_f = 0.02 \times 75000 \text{ N} = 1500 \text{ N} \]
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Calculate the angular velocity (\( \omega \)): The angular velocity in radians per second can be calculated from the rotational speed: \[ \text{Rotational speed} = 350 \text{ r/min} \] To convert this to radians per second, we use the following relation: \[ \omega = \text{Rotational speed} \times \frac{2\pi \text{ rad}}{1 \text{ rev}} \times \frac{1 \text{ min}}{60 \text{ s}} \] Plugging in the values: \[ \omega = 350 \times \frac{2\pi}{60} \approx 36.65 \text{ rad/s} \]
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Calculate the power lost due to friction: The power lost due to friction can be calculated using the formula: \[ P = F_f \cdot v \] where \( v \) is the linear speed at the radius of the shaft, calculated using: \[ v = r \cdot \omega \]
First, calculate \( v \): \[ v = 0.125 \text{ m} \times 36.65 \text{ rad/s} \approx 4.58125 \text{ m/s} \]
Now, substituting \( v \) into the power equation: \[ P = 1500 \text{ N} \cdot 4.58125 \text{ m/s} \approx 6871.88 \text{ W} \]
To convert watts to kilowatts: \[ P \approx 6.87 \text{ kW} \]
Thus, the power lost due to friction is approximately 6.87 kW.
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