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)

0.5 m

Answer 2

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 $0.5\, m$.