Two stones of masses $m$ and $3 m$ are whirled in horizontal circles, the heavier one in a radius $r

Question

Two stones of masses $m$ and $3 m$ are whirled in horizontal circles, the heavier one in a radius $r / 3$ and the lighter one in radius $r$. The tangential speed of lighter stone is $n$ times that of the value of heavier stone when they experience same centripetal forces. The value of $n$ is\

Professor PC Thomas Classes · Answer

Let $v$ be tangential speed of heavier stone. Then, centripetal force experienced by lighter stone is\ $ \left(F_{c}ight)_{	ext {lighter }}=\frac{m(n v)^{2}}{r} $ and that of heavier stone is $\left(F_{c}ight)_{	ext {heavier }}=\frac{3 m v^{2}}{(r / 3)}$\ $\operatorname{But}\left(F_{c}ight)_{	ext {lighter }}=\left(F_{c}ight)_{	ext {heavier }} \quad$ (given)\ $	herefore \quad \frac{m(n v)^{2}}{r}=\frac{3 m v^{2}}{(r / 3)}$ or, $n^{2}\left(\frac{m v^{2}}{r}ight)=9\left(\frac{m v^{2}}{r}ight)$ $$ n^{2}=9 \quad 	ext { or } \quad n=3 $$