This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
ScanSolve AI Answer
You're on a roll — Step 1: Identify the given radii of the two initial soap bubbles. Let R_1 be the radius of the first bubble and R_2 be the radius of the second bubble. R_1 = 3 cm R_2 = 6 cm Step 2: Recall the physical principles for coalescing soap bubbles in a vacuum under isothermal conditions. For a soap bubble, the excess pressure inside is given by P = (4T)/(R), where T is the surface tension and R is the radius. Since the bubbles are in a vacuum, the external pressure is 0, so the absolute pressure inside the bubble is P = (4T)/(R). The volume of a spherical bubble is V = (4)/(3) R^3. When two bubbles coalesce under isothermal conditions, the total number of moles of gas inside is conserved. From the ideal gas law (PV = nRT), since T is constant, n PV. Therefore, the product PV is conserved: P_1 V_1 + P_2 V_2 = P_f V_f where P_f and V_f are the pressure and volume of the final bubble. Step 3: Substitute the expressions for pressure and volume into the conservation equation. ((4T)/(R_1)) ((4)/(3) R_1^3) + ((4T)/(R_2)) ((4)/(3) R_2^3) = ((4T)/(R_f)) ((4)/(3) R_f^3) Cancel the common factor ((4T · 4)/(3)) from all terms: (R_1^3)/(R_1) + (R_2^3)/(R_2) = (R_f^3)/(R_f) R_1^2 + R_2^2 = R_f^2 Step 4: Calculate the radius of the final bubble (R_f). Substitute the given values for R_1 and R_2: R_f^2 = (3 cm)^2 + (6 cm)^2 R_f^2 = 9 cm^2 + 36 cm^2 R_f^2 = 45 cm^2 R_f = sqrt(45 cm)^2 R_f = sqrt(9 × 5) cm R_f = 3sqrt(5) cm The numerical value is approximately: R_f ≈ 3 × 2.236 cm R_f ≈ 6.708 cm The radius of the bubble formed is 3sqrt(5) cm. Send me the next one 📸