Question

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

• A. 1 m
• B. 2 m
• C. 0.5 m
• D. 4 m

Answer 1

Answer: (C)

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 $2s$, then it is called a second's pendulum.
Also, $g=\frac{GM}{R^2}$
where, $M$ is mass, $R$ is radius
$\therefore T=2\pi \sqrt{\frac{R^{2}l}{GM}}=2\dots$(1)
Second's pendulum on other planet is
$2=2\pi \sqrt{\frac{4R^2{l}^{\prime }}{G(2M)}}\dots$(2)
From Eqs. (1) and (2), we have
$\frac{R^{2}l}{GM}=\frac{4R^{2}l}{G(2M)}$
$\Rightarrow l^{\prime }=0.5m$
Hence, length of pendulum on planet is $0.5m$.