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Q.
The length of second's pendulum is 1 m on earth. If mass and diameter of a planet is doubled than that of earth, then its length becomes
Punjab PMETPunjab PMET 2006Oscillations
Solution:
The motion of the bob is simple harmonic, hence its time period is given by
$T=2 \pi \sqrt{\frac{\text { displacement }}{\text { acceleration }}}=2 \pi \sqrt{\frac{l}{g}}$
Also if the periodic time of a pendulum is $2 s$, then it is called a second's pendulum.
Also, $g=\frac{G M}{R^{2}}$
where, $M$ is mass, $R$ is radius
$\therefore T=2 \pi \sqrt{\frac{R^{2} l}{G M}}=2 \ldots$(1)
Second's pendulum on other planet is
$2=2 \pi \sqrt{\frac{4 R^{2} l^{\prime}}{G(2 M)}} \ldots$(2)
From Eqs. (1) and (2), we have
$ \frac{R^{2} l}{G M} =\frac{4 R^{2} l}{G(2 M)} $
$\Rightarrow l' =0.5\, m$
Hence, length of pendulum on planet is $0.5\, m$.