To calculate the power lost due to friction in a bearing where a shaft is rotating, we can use the following formula:
\[ P = F \cdot v \]
Where:
- \( P \) is the power lost due to friction (in watts),
- \( F \) is the frictional force (in newtons), and
- \( v \) is the velocity (in meters per second) at the point of contact.
Step 1: Calculate the Frictional Force
The frictional force \( F \) can be calculated using the formula:
\[ F = \mu \cdot N \]
Where:
- \( \mu \) is the coefficient of friction (given as 0.03),
- \( N \) is the normal force (which is equal to the axial force exerted on the bearing, given as 70 kN or 70000 N).
Substituting the values:
\[ F = 0.03 \cdot 70000 = 2100 \text{ N} \]
Step 2: Calculate the Velocity
The velocity at the surface of the shaft can be calculated as follows:
- First, we need to find the circumference of the shaft:
The diameter of the shaft (\( d \)) = 200 mm = 0.2 m
The radius (\( r \)) = \( \frac{d}{2} \) = 0.1 m.
The circumference (\( C \)) is given by:
\[ C = 2 \pi r = 2 \pi (0.1) \approx 0.6283 \text{ m} \]
- Convert the shaft rotation speed from revolutions per minute (r/min) to radians per second:
The shaft rotates at 300 r/min. To convert this to revolutions per second:
\[ \text{Revolutions per second} = \frac{300}{60} = 5 \text{ r/s} \]
Now, convert revolutions to radians:
\[ \text{Angular velocity} (ω) = 5 \text{ r/s} \cdot 2\pi \text{ rad/rev} \approx 31.42 \text{ rad/s} \]
- Finally, calculate the linear velocity \( v \):
\[ v = r \cdot ω = 0.1 \cdot 31.42 \approx 3.142 \text{ m/s} \]
Step 3: Calculate the Power Lost Due to Friction
Using the power formula:
\[ P = F \cdot v = 2100 \text{ N} \cdot 3.142 \text{ m/s} \approx 6598.2 \text{ W} \]
Conclusion
The power lost due to friction is approximately \( 6598.2 \text{ W} \), or about \( 6.6 \text{ kW} \).