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Question

A solid cube floats in water half immersed and has small vertical oscillations of time period π5s. Its mass (in kg) is:
(Take g=10 ms2)

A
4
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B
2
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C
1
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D
0.5
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Solution

The correct option is A 4
Given,
Time period of oscillation (T)=π5s
Density of liquid (ρ)=1000kg/m3
Since, cube is half immersed i.e h=l2 is the height of cube exposed above water surface.
Applying the formula for floatation,
hl=σρ
where, l edge length of with cubical block
σ Density of block
ρ density of liquid

l2×l=σρ
σ=ρ2
i.e density of cube is half the density of liquid
σ=500 kg/m3
Time period of oscillations is given as,
T=2πmρAg ....(i)
Where m=σl3=mass of cubical block
A=l2=cross-sectional area of cubical block
Eq.(i) can be expressed as,
T=2πσlρg
Substituting σρ=12
π5=2πl2g
l20=1100
(or)l=0.2 m
Now mass of cube is,
m=V×σ=l3σ=(0.2)3×500
m=4 kg

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