Oscillations

JEE Main level, single correct, medium

129 of 194

Question 129 of 194

A simple pendulum oscillating in air has period TT . The bob of the pendulum is completely immersed in a non-viscous liquid. The density of the liquid is 116th\dfrac{1}{16}th of the material of the bob. If the bob is inside liquid all the time, its period of oscillation in this liquid is:

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Answer: (C) 4T1154T\sqrt{\dfrac{1}{15}}

Let the density of bob is σ\sigma & density of liquid is σ16\dfrac{\sigma }{16} .
Now, Net downward force on bob = Weight - Buoyancy
=σVg−σ16Vg=1516σVg=\sigma Vg-\dfrac{\sigma }{16}Vg=\dfrac{15}{16}\sigma Vg
Net effective gravitational acceleration =1516σVgσV=1516g=\dfrac{\dfrac{15}{16}\sigma Vg}{\sigma V}=\dfrac{15}{16}g
Time period (T)=2πlgeff=2πl1516g=2π1615lg=1615∙(2πlg)=415T\left(T\right)=2\pi \sqrt{\dfrac{l}{{g}_{eff}}}=2\pi \sqrt{\dfrac{l}{\dfrac{15}{16}g}}=2\pi \sqrt{\dfrac{16}{15}\dfrac{l}{g}}=\sqrt{\dfrac{16}{15}}∙\left(2\pi \sqrt{\dfrac{l}{g}}\right)=\dfrac{4}{\sqrt{15}}T
=4T115=4T\sqrt{\dfrac{1}{15}}

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