The inner race of a bearing rotates steadily with an angular velocity of 1,393 radians per second. The bearing is lubricated by an oil with viscosity 0.04 Pa*s. The oil is retained between the inner race and the outer race of the bearing, with a gap of 0.08 mm. The diameter of the inner race is 7.7 cm and the axial length of the bearing is 1 cm.

Determine the torque in N*m required to maintain the angular velocity of the inner race, if the outer race is unmoving.



To find the torque required to maintain the angular velocity of the inner race, we need to analyze the viscous forces acting on the oil film between the inner and outer races. Here's how we can proceed with the calculations:

### Step-by-step Solution

1. **Given Data:**
   - Angular velocity of inner race, \(\omega = 1,393 \, \text{rad/s}\)
   - Viscosity of the oil, \(\mu = 0.04 \, \text{Pa} \cdot \text{s}\)
   - Gap between the inner and outer race, \(h = 0.08 \, \text{mm} = 0.00008 \, \text{m}\)
   - Diameter of the inner race, \(D = 7.7 \, \text{cm} = 0.077 \, \text{m}\)
   - Axial length of the bearing, \(L = 1 \, \text{cm} = 0.01 \, \text{m}\)

2. **Velocity gradient:**
   The fluid between the inner and outer races experiences shear due to the movement of the inner race. Assuming a linear velocity profile:
   \[
   \frac{d u}{d y} = \frac{\omega R}{h}
   \]
   where \(R = \frac{D}{2} = \frac{0.077}{2} = 0.0385 \, \text{m}\).

   Substitute the values:
   \[
   \frac{d u}{d y} = \frac{1,393 \times 0.0385}{0.00008} \approx 670,393.75 \, \text{s}^{-1}.
   \]

3. **Shear stress in the fluid:**
   The shear stress, \(\tau\), in the fluid due to the viscosity is given by:
   \[
   \tau = \mu \frac{d u}{d y}.
   \]
   Substitute the values:
   \[
   \tau = 0.04 \times 670,393.75 \approx 26,815.75 \, \text{Pa}.
   \]

4. **Force acting on the fluid layer:**
   The shear force \(F\) acting on the fluid layer is the product of shear stress and the surface area \(A\) of the inner race:
   \[
   A = 2 \pi R L = 2 \pi \times 0.0385 \times 0.01 \approx 0.002418 \, \text{m}^2.
   \]
   \[
   F = \tau A = 26,815.75 \times 0.002418 \approx 64.78 \, \text{N}.
   \]

5. **Torque required:**
   Torque \(T\) is the product of the force and the radius:
   \[
   T = F R = 64.78 \times 0.0385 \approx 2.49 \, \text{N} \cdot \text{m}.
   \]

So, the torque required to maintain the angular velocity of the inner race is approximately **2.49 N·m**.

Do you need further details or have any questions?

### Relative Questions:
1. How does the viscosity of the lubricant affect the torque required in such a system?
2. What would happen to the required torque if the gap between the races increased?
3. How would the torque change if the diameter of the inner race were larger?
4. What is the significance of the linear velocity profile assumption in this calculation?
5. How would you account for temperature changes affecting the viscosity in a real-world application?

### Tip:
Always ensure the units are consistent throughout the calculations to avoid errors, especially when dealing with small quantities like the gap in lubrication problems.

The inner race of a bearing rotates steadily with an angular velocity of 1,393 radians per second. The bearing is lubricated by an oil with viscosity 0.04 Pa*s. The oil is retained between the inner race and the outer race of the bearing, with a gap of 0.08 mm. The diameter of the inner race is 7.7 cm and the axial length of the bearing is 1 cm. Determine the torque in N*m required to maintain the angular velocity of the inner race, if the outer race is unmoving.

Determine the torque required to maintain an angular velocity of 1,393 rad/s in a bearing lubricated by oil with 0.04 Pa·s viscosity. This step-by-step guide covers torque calculation using principles of fluid mechanics and viscous forces. Suitable for undergraduate mechanical engineering students.

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