In the previous problem, if 15.0 cm of water and spirit each are further poured into the respective arms of the tube, what is the difference in the levels of mercury in the two arms? (Specific gravity of mercury = 13.6)
The difference in the levels of mercury in the two arms is approximately 0.22 cm (or 0.221 cm), with mercury rising higher in the spirit arm.
Explanation
This question builds upon the previous problem (9.9) where we determined the specific gravity of spirit. From the initial equilibrium condition with equal mercury levels, we found that specific gravity of spirit = 0.8. When additional liquids are poured, the pressure balance shifts, causing mercury to rise in the spirit arm where the effective pressure from the liquid column is lower. The pressure balance equation at the mercury surface level gives us the difference in mercury heights.
Solution Steps
Step 1: Find specific gravity of spirit from previous problem From problem 9.9, mercury levels were equal with 10.0 cm water and 12.5 cm spirit. At equilibrium: ρ_water × g × 10.0 = ρ_spirit × g × 12.5 Specific gravity of spirit = ρ_spirit/ρ_water = 10.0/12.5 = 0.8
Step 2: Calculate new heights after adding liquids Water column: 10.0 + 15.0 = 25.0 cm Spirit column: 12.5 + 15.0 = 27.5 cm
Step 3: Apply pressure balance at mercury surface Let Δh be the difference in mercury levels (mercury higher in spirit arm). At mercury surface in water arm: Pressure = P_atm + ρ_w × g × 25.0 At same level in spirit arm: Pressure = P_atm + ρ_s × g × 27.5 + ρ_m × g × Δh For equilibrium: ρ_w × 25.0 = ρ_s × 27.5 + ρ_m × Δh
Step 4: Solve for Δh Using ρ_s = 0.8 × ρ_w and ρ_m = 13.6 × ρ_w: 25.0 = + 13.6 × Δh 25.0 = 22.0 + 13.6 × Δh Δh = 3.0/13.6 = 0.221 cm 0.22 cm