A column of mercury 10 cm long is contained in the middle of a narrow, horizontal 1 m long tube which is closed at both ends. Both the halves of the tube contain air at a pressure of 76 cm of mercury. By what distance in centimeter (approximately) will the column of mercury be displaced if the tube is held vertically?
Physics · Mixed Chapter
JEE Mains

Question

A column of mercury 10 cm long is contained in the middle of a narrow, horizontal 1 m long tube which is closed at both ends. Both the halves of the tube contain air at a pressure of 76 cm of mercury. By what distance in centimeter (approximately) will the column of mercury be displaced if the tube is held vertically?

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1: Initially each air column is (100−10)/2=45 cm at pressure 76 cm Hg. Let the mercury plug move distance x downwards on turning the tube vertical. Step 2: By Boyle’s law, lower and upper gas pressures are P_l=76×45/(45−x) and P_u=76×45/(45+x), in cm Hg. Step 3: Their pressure difference balances 10 cm of mercury: P_l−P_u=10. Therefore 76×45×[2x/(45²−x²)]=10. Step 4: Solve for the small positive root: x≈2.95 cm, i.e. approximately 3 cm.