Let's break down the problem from the image: **Question:** Determine the Sommerfeld number of a bearing whose clearance ratio is 600 and the bearing load is 750 lbf. The length and radius of the given bearing is 1.75 in. and 0.75 in. respectively. The viscosity of the oil inside the bearing is μ = 6 cP (centipoise). The bearing is rotating at a speed of 1800 rpm. --- ### Given Data: - Clearance ratio, \( r/c = 600 \) - Bearing load, \( W = 750 \) lbf - Length of bearing, \( L = 1.75 \) in - Radius of bearing, \( r = 0.75 \) in - Viscosity, \( \mu = 6 \) cP = \( 6 \times 10^{-3} \) Pa.s (since \( 1 \) cP \( = 10^{-3} \) Pa.s) - Speed, \( N = 1800 \) rpm --- ### Step 1: Sommerfeld Number Formula \[ S = \frac{(r/c)^2 \cdot (\mu N)}{P} \] But commonly used formula in English units is: \[ S = \frac{(r/c)^2 \cdot (\mu N r)}{P} \] Where: - \( \mu \) in lbf·s/in² - \( N \) in rev/s - \( r \) in inches - \( P \) is the bearing pressure in lbf/in² #### Bearing Pressure \( P \): \[ P = \frac{W}{L \cdot 2r} \] --- ### Step 2: Convert Units - Viscosity: \( 6 \) cP = \( 6 \times 10^{-3} \) Pa·s - 1 Pa·s = 0.020885 lbf·s/in² - \( 6 \times 10^{-3} \times 0.020885 = 1.2531 \times 10^{-4} \) lbf·s/in² - Speed: \( 1800 \) rpm = \( 30 \) rev/s --- ### Step 3: Calculate Clearance \( c \) \[ r/c = 600 \implies c = \frac{r}{600} = \frac{0.75}{600} = 0.00125 \text{ in} \] --- ### Step 4: Calculate Bearing Pressure \[ P = \frac{W}{L \cdot 2r} = \frac{750}{1.75 \times 2 \times 0.75} = \frac{750}{2.625} = 285.71 \text{ lbf/in}^2 \] --- ### Step 5: Plug into Sommerfeld Number Formula \[ S = \frac{(r/c)^2 \cdot (\mu N r)}{P} \] Plugging in values: - \( r/c = 600 \) - \( \mu = 1.2531 \times 10^{-4} \) lbf·s/in² - \( N = 30 \) rev/s - \( r = 0.75 \) in - \( P = 285.71 \) lbf/in² \[ S = \frac{(600)^2 \times (1.2531 \times 10^{-4}) \times 30 \times 0.75}{285.71} \] \[ = \frac{360,000 \times 1.2531 \times 10^{-4} \times 22.5}{285.71} \] \[ = \frac{360,000 \times 0.002819475}{285.71} \] \[ = \frac{1015.011}{285.71} \] \[ = 0.355 \] --- ### Step 6: Closest Answer Looking at the options (0.58, 0.36, 0.28, 0.23), the closest value is **0.36**. --- **Final Answer:** \[ \boxed{0.36} \]