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Question

A uniform candle of length 10 cm is glued with an aluminium cylinder of same radius and length 1 cm. Composite system is submerged in the water and it floats stably in vertical orientation. Density of wax is 0.8 g/cm3, density of aluminium is 2.7 g/cm3 and density of water is 1.0 g/cm3. At t=0 candle is lit and assume that it consumes wax at a constant rate of 0.1 cm/min. List -1 gives four different instant of time while List-ll lists magnitude of some quantity. List-IList-II(i)t=5 minutes(P)103107(ii)t=10 minutes(Q)9.3(iii)t=15 minutes(R)8.9(iv)t=20 minutes(S)99107(T)8.5(U)95107
If candle is extinguished at different times i.e. at t=5 minute,t=10 minute,t=15 minute and at
t=20 minutes and slightly displaced from mean position in each case. The correct match for time
period of small oscillation in each case will be (in units of T0 where T0 is time period at t=0)

A
i P, ii P, iii U, iv U
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B
i Q, ii U, iii T, iv T
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C
i Q, ii P, iii S, iv S
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D
i P, ii S, iii U, iv U
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Solution

The correct option is D i P, ii S, iii U, iv U
l0A×2.7+lA×0.8=(l0+y)A
l0(2.71)+0.8l=y
y=1.7l0+0.8l
y=1.7+0.8l
at t=5 minute;l=9.5 cm, y=9.3
at t=10 minute;l=9 cm, y=8.9
at t=15 minute;l=8.5 cm, y=8.5
at t=20 minute;l=8 cm, y=8.1 which is not possible y=8.5
T=2πmAρg
m0=k(2.7+8)=10.7k
m5=k(2.7+7.6)=10.3k
m10=k(2.7+7.2)=9.9k
m15=k(2.7+6.8)=9.5k
m20=k(2.7+6.8)=9.5k
T010.7
T510.3103107T0
T10=99107T0
T15=95107T0
T20=95107T0

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