Figure-shows-a-small-bob-of-mass-m-suspended-from-a-point-on-a-thin-rod-by-a-light-inextensible-string-of-length-l-The-rod-is-rigidly-fixed-on-a-circular-platform-The-platform-is-set-into-rotation-

Question Number 21131 by Tinkutara last updated on 13/Sep/17

$$\mathrm{Figure}\:\mathrm{shows}\:\mathrm{a}\:\mathrm{small}\:\mathrm{bob}\:\mathrm{of}\:\mathrm{mass}\:{m} \\ $$$$\mathrm{suspended}\:\mathrm{from}\:\mathrm{a}\:\mathrm{point}\:\mathrm{on}\:\mathrm{a}\:\mathrm{thin}\:\mathrm{rod} \\ $$$$\mathrm{by}\:\mathrm{a}\:\mathrm{light}\:\mathrm{inextensible}\:\mathrm{string}\:\mathrm{of}\:\mathrm{length} \\ $$$${l}.\:\mathrm{The}\:\mathrm{rod}\:\mathrm{is}\:\mathrm{rigidly}\:\mathrm{fixed}\:\mathrm{on}\:\mathrm{a}\:\mathrm{circular} \\ $$$$\mathrm{platform}.\:\mathrm{The}\:\mathrm{platform}\:\mathrm{is}\:\mathrm{set}\:\mathrm{into} \\ $$$$\mathrm{rotation}.\:\mathrm{The}\:\mathrm{minimum}\:\mathrm{angular}\:\mathrm{speed} \\ $$$$\omega,\:\mathrm{for}\:\mathrm{which}\:\mathrm{the}\:\mathrm{bob}\:\mathrm{loses}\:\mathrm{contact}\:\mathrm{with} \\ $$$$\mathrm{the}\:\mathrm{vertical}\:\mathrm{rod},\:\mathrm{is} \\ $$$$\left(\mathrm{1}\right)\:\sqrt{\frac{{g}}{{l}}} \\ $$$$\left(\mathrm{2}\right)\:\sqrt{\frac{\mathrm{2}{g}}{{l}}} \\ $$$$\left(\mathrm{3}\right)\:\sqrt{\frac{{g}}{\mathrm{2}{l}}} \\ $$$$\left(\mathrm{4}\right)\:\sqrt{\frac{{g}}{\mathrm{4}{l}}} \\ $$

Commented by ajfour last updated on 13/Sep/17

sin θ ≈ θ =(r/l) Tcos θ ≈ T=mg Tsin θ ≈ Tθ = T ((r/l))=mω^2 r ⇒ mg((r/l))=mω^2 r or ω=(√(g/l)) . option (1).

$$\mathrm{sin}\:\theta\:\approx\:\theta\:=\frac{{r}}{{l}} \\ $$$${T}\mathrm{cos}\:\theta\:\approx\:{T}={mg} \\ $$$${T}\mathrm{sin}\:\theta\:\approx\:{T}\theta\:=\:{T}\:\left(\frac{{r}}{{l}}\right)={m}\omega^{\mathrm{2}} {r} \\ $$$$\Rightarrow\:\:\:\:{mg}\left(\frac{{r}}{{l}}\right)={m}\omega^{\mathrm{2}} {r} \\ $$$${or}\:\:\:\omega=\sqrt{\frac{{g}}{{l}}}\:.\:\:\:{option}\:\left(\mathrm{1}\right). \\ $$

Commented by ajfour last updated on 13/Sep/17