Question

A shaft 250 mm in diameter is revolving and exerting a force of 75 kN on a bearing.
Calculate the power which is lost due to friction if the coefficient of friction between the shaft and the bearing is 0,02, and the shaft rotates at 350 r/min.

Answer 1

To calculate the power lost due to friction in the bearing, we can follow these steps:

1. **Calculate the radius of the shaft**:
\[
\text{Diameter} = 250 \text{ mm} = 0.250 \text{ m}
\]
\[
\text{Radius} (r) = \frac{\text{Diameter}}{2} = \frac{0.250 \text{ m}}{2} = 0.125 \text{ m}
\]

2. **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:
\[
F_f = 0.02 \times 75000 \text{ N} = 1500 \text{ N}
\]

3. **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:
\[
\omega = 350 \times \frac{2\pi}{60} \approx 36.65 \text{ rad/s}
\]

4. **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**.