Question

# A cylinder of height 1 m is floating in water at 0∘C with 20 cm height in air. When the temperature of water is raised to 4∘C, height of the cylinder in air becomes 21 cm. By assuming the expansion of the cylinder to be negligible and that the cylinder stays vertical, find the ratio of density of water at 4∘C to density of water at 0∘C.

A
1.01
B
1.03
C
2.01
D
1.04

Solution

## The correct option is A 1.01Given, height of cylinder (h)=1 m=100 cm From the data given in the question we can say that, Length of cylinder submerged in water at 0∘C is l1=80 cm Length of cylinder submerged in water at 4∘C is l2=79 cm Pictorial representation of situation given in the question is shown below. From Archimedes principle, we can say that Buoyant force = weight of the fluid displaced. i.e Fb=Vfρfg and for vertical equilibrium of cylinder, weight mg=Vsρsg=Fb=Vfρfg At 0∘C: Vsρsg=Vfρfg ⇒ρsρf=VfVs=Height of cylinder immersedTotal height of cylinder ⇒ρsρf=l1h=80100 ....(1) Similarly, at 4∘C: Vsρsg=V′fρ′fg where ρ′f is the new density of fluid and V′f is the new volume displaced ⇒ρsρ′f=Height of cylinder immersedTotal height of cylinder ⇒ρsρ′f=l2h=79100    ...........(2) So, from (1) and (2) ρ′fρf=8079=1.01 Thus, option (a) is the correct answer.

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