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Question

A cylinder of height 1 m is floating in water at 0C with 20 cm height in air. When the temperature of water is raised to 4C, 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 4C to density of water at 0C


Correct Answer
A
1.01
Your Answer
B
1.03
Your Answer
C
2.01
Your Answer
D
1.04

Solution

The correct option is A 1.01
Given, 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 0C is l1=80 cm
Length of cylinder submerged in water at 4C 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 0C:
Vsρsg=Vfρfg
ρsρf=VfVs=Height of cylinder immersedTotal height of cylinder
ρsρf=l1h=80100 ....(1)

Similarly, at 4C:
Vsρsg=Vfρfg
where ρf is the new density of fluid and Vf is the new volume displaced
ρsρf=Height of cylinder immersedTotal height of cylinder
ρsρf=l2h=79100    ...........(2)
So, from (1) and (2)
ρfρf=8079=1.01
Thus, option (a) is the correct answer.

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