A small hole is made at the bottom of a symmetrical jar as shown in fig. A liquid is filled into the jar up to a certain height. The rate of fall of liquid surface is independent of the level of liquid in the jar. Then the surface of jar is a surface of revolution of the curve y=kxⁿ, the value of n is
Physics · Mixed Chapter
JEE Mains

Question

A small hole is made at the bottom of a symmetrical jar as shown in fig. A liquid is filled into the jar up to a certain height. The rate of fall of liquid surface is independent of the level of liquid in the jar. Then the surface of jar is a surface of revolution of the curve y=kxⁿ, the value of n is

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1: Let a be the fixed area of the small outlet. For liquid height y, Torricelli efflux rate is Qₒᵤₜ=a√(2gy), proportional to √y. Step 2: The volume decrease rate is also A(y)|dy/dt|, where A(y)=πx(y)² is the horizontal liquid-surface area. Step 3: If the falling level speed |dy/dt| is constant, continuity requires A(y)∝√y. Thus x²∝y¹ᐟ². Step 4: Squaring gives x⁴∝y or y=kx⁴. Hence n=4.