A journal bearing of radius $$‘r_1’$$​ and radial clearance ‘c1’ has a sommerfeld number $$‘s_1’$$​. Another journal bearing of radius $$‘r_2’$$​ and radial clearance $$‘c_2’$$ has a sommerfeld number $$‘s_2’$$​. All the other parameters remain the same for both bearings. If $$r_2$$​= 2 $$r_1$$​ and $$c_2$$ = $$c_1$$​/2, then $$s_2/s_1$$​​

Correct option is C

​$$\textbf{Sommerfeld Number Formula:} \\S = \left( \frac{r}{c} \right)^2 \cdot \frac{\mu N}{P} \\\\\text{Where:} \\\bullet \ r = \text{journal radius} \\\bullet \ c = \text{radial clearance} \\\bullet \ \mu = \text{dynamic viscosity} \\\bullet \ N = \text{rotational speed} \\\bullet \ P = \text{unit load (pressure)} \\\\\text{Since } \mu, N, P \text{ remain constant for both bearings, the Sommerfeld number is proportional to } \left( \frac{r}{c} \right)^2. \\\\\textbf{Given:} \\r_2 = 2r_1 \\c_2 = \frac{c_1}{2} \\\\\text{So,} \\\frac{s_2}{s_1} = \left( \frac{r_2 / c_2}{r_1 / c_1} \right)^2 = \left( \frac{2r_1 / (c_1 / 2)}{r_1 / c_1} \right)^2 = \left( \frac{2r_1 \cdot 2}{r_1} \right)^2 = (4)^2 = 16$$​​