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Oscillations — Physics STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 SciencePhysicsOscillations5 Marks
Question
Two pendulums of lengths 100cm and 110.25cm start oscillating in phase simultaneously. After how many oscillations will they again be in phase together?
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Answer

$\text{T}=2\pi\sqrt{\frac{\text{l}}{\text{g}}}$$\text{l}_1=100\text{cm}$
$\text{l}_2=110.25\text{cm}$
For smaller pendulum, $\text{T}_1=2\pi\sqrt{\frac{100}{\text{g}}}\cdots\text{(i)}$
For larger pendulum, $\text{T}_2=2\pi\sqrt{\frac{110.25}{\text{g}}}\cdots\text{(ii)}$
Let these pendulums oscillate in phase again if larger pendulum completes 'n' oscillations. It means smaller pendulum must complete (n + 1) oscillations.
$\text{nT}_2=(\text{n}+1)\text{T}_1$
$\frac{\text{n}+1}{\text{n}}$
$\frac{\text{T}_2}{\text{T}_1}=\sqrt{\frac{110.25}{100}}$
$=1.05$
$=1+\frac{1}{\text{n}}$
$=1.05$
$=\frac{1}{\text{n}}=0.05$
$=\frac{5}{100}=\frac{1}{20}$
$\therefore\text{n}=20$
Hence both pendulums will again oscillate in phase after 20 oscillations of the larger or 21 oscillations of the smaller pendulum.

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