Question

The length of a second pendulum on earth is $\ell$. The length of a seconds pendulum on a planet the mass and radius of which are twice those of the earth will be:

• A. $\ell$
• B. $\ell /\sqrt{2}$
• C. $\ell /2$
• D. $\sqrt{2\ell }$

Answer 1

The correct option is C $\ell /2$

For a pendulum of length $l$ the time period is $T=2\pi \sqrt[2]{2l/g}$

Now the time period is needed to be constant because we want second pendulum again.

By changing the planet we are merely changing the value of $g$

On earth the $g=GM/R^2$

for new planet just replace the mass $M$ by $2M$ and $R$ by $2R$

thus new value of g is $g\prime =2GM/4R^2=GM/2R^2=g/2$

As value of $g$ became $half$ of initial value so to maintain the ratio $l/g$ constant we need $half$ length i.e. $l/2$

Option C is correct.