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Consider-a-disc-rotating-in-the-horizontal-plane-with-a-constant-angular-speed-about-its-centre-O-The-disc-has-a-shaded-region-on-one-side-of-the-diameter-and-an-unshaded-region-on-the-other-side-a




Question Number 20745 by Tinkutara last updated on 02/Sep/17
Consider a disc rotating in the horizontal  plane with a constant angular speed ω  about its centre O. The disc has a  shaded region on one side of the diameter  and an unshaded region on the other  side as shown in the Figure. When the  disc is in the orientation as shown, two  pebbles P and Q are simultaneously  projected at an angle towards R. The  velocity of projection is in the y-z  plane and is same for both pebbles with  respect to the disc. Assume that (i) they  land back on the disc before the disc has  completed (1/8) rotation, (ii) their range  is less than half the disc radius, and  (iii) ω remains constant throughout.  Then  (a) P lands in the shaded region and Q  in the unshaded region  (b) P lands in the unshaded region and  Q in the shaded region  (c) Both P and Q land in the unshaded  region  (d) Both P and Q land in the shaded  region
$$\mathrm{Consider}\:\mathrm{a}\:\mathrm{disc}\:\mathrm{rotating}\:\mathrm{in}\:\mathrm{the}\:\mathrm{horizontal} \\ $$$$\mathrm{plane}\:\mathrm{with}\:\mathrm{a}\:\mathrm{constant}\:\mathrm{angular}\:\mathrm{speed}\:\omega \\ $$$$\mathrm{about}\:\mathrm{its}\:\mathrm{centre}\:{O}.\:\mathrm{The}\:\mathrm{disc}\:\mathrm{has}\:\mathrm{a} \\ $$$$\mathrm{shaded}\:\mathrm{region}\:\mathrm{on}\:\mathrm{one}\:\mathrm{side}\:\mathrm{of}\:\mathrm{the}\:\mathrm{diameter} \\ $$$$\mathrm{and}\:\mathrm{an}\:\mathrm{unshaded}\:\mathrm{region}\:\mathrm{on}\:\mathrm{the}\:\mathrm{other} \\ $$$$\mathrm{side}\:\mathrm{as}\:\mathrm{shown}\:\mathrm{in}\:\mathrm{the}\:\mathrm{Figure}.\:\mathrm{When}\:\mathrm{the} \\ $$$$\mathrm{disc}\:\mathrm{is}\:\mathrm{in}\:\mathrm{the}\:\mathrm{orientation}\:\mathrm{as}\:\mathrm{shown},\:\mathrm{two} \\ $$$$\mathrm{pebbles}\:{P}\:\mathrm{and}\:{Q}\:\mathrm{are}\:\mathrm{simultaneously} \\ $$$$\mathrm{projected}\:\mathrm{at}\:\mathrm{an}\:\mathrm{angle}\:\mathrm{towards}\:{R}.\:\mathrm{The} \\ $$$$\mathrm{velocity}\:\mathrm{of}\:\mathrm{projection}\:\mathrm{is}\:\mathrm{in}\:\mathrm{the}\:{y}-{z} \\ $$$$\mathrm{plane}\:\mathrm{and}\:\mathrm{is}\:\mathrm{same}\:\mathrm{for}\:\mathrm{both}\:\mathrm{pebbles}\:\mathrm{with} \\ $$$$\mathrm{respect}\:\mathrm{to}\:\mathrm{the}\:\mathrm{disc}.\:\mathrm{Assume}\:\mathrm{that}\:\left({i}\right)\:\mathrm{they} \\ $$$$\mathrm{land}\:\mathrm{back}\:\mathrm{on}\:\mathrm{the}\:\mathrm{disc}\:\mathrm{before}\:\mathrm{the}\:\mathrm{disc}\:\mathrm{has} \\ $$$$\mathrm{completed}\:\frac{\mathrm{1}}{\mathrm{8}}\:\mathrm{rotation},\:\left({ii}\right)\:\mathrm{their}\:\mathrm{range} \\ $$$$\mathrm{is}\:\mathrm{less}\:\mathrm{than}\:\mathrm{half}\:\mathrm{the}\:\mathrm{disc}\:\mathrm{radius},\:\mathrm{and} \\ $$$$\left({iii}\right)\:\omega\:\mathrm{remains}\:\mathrm{constant}\:\mathrm{throughout}. \\ $$$$\mathrm{Then} \\ $$$$\left({a}\right)\:{P}\:\mathrm{lands}\:\mathrm{in}\:\mathrm{the}\:\mathrm{shaded}\:\mathrm{region}\:\mathrm{and}\:{Q} \\ $$$$\mathrm{in}\:\mathrm{the}\:\mathrm{unshaded}\:\mathrm{region} \\ $$$$\left({b}\right)\:{P}\:\mathrm{lands}\:\mathrm{in}\:\mathrm{the}\:\mathrm{unshaded}\:\mathrm{region}\:\mathrm{and} \\ $$$${Q}\:\mathrm{in}\:\mathrm{the}\:\mathrm{shaded}\:\mathrm{region} \\ $$$$\left({c}\right)\:\mathrm{Both}\:{P}\:\mathrm{and}\:{Q}\:\mathrm{land}\:\mathrm{in}\:\mathrm{the}\:\mathrm{unshaded} \\ $$$$\mathrm{region} \\ $$$$\left({d}\right)\:\mathrm{Both}\:{P}\:\mathrm{and}\:{Q}\:\mathrm{land}\:\mathrm{in}\:\mathrm{the}\:\mathrm{shaded} \\ $$$$\mathrm{region} \\ $$
Commented by Tinkutara last updated on 02/Sep/17
Commented by NECx last updated on 02/Sep/17
how did you draw this?
$${how}\:{did}\:{you}\:{draw}\:{this}? \\ $$
Commented by Tinkutara last updated on 02/Sep/17
I didn′t drew this; I took screenshot of  this figure from a typed question paper  PDF.
$$\mathrm{I}\:\mathrm{didn}'\mathrm{t}\:\mathrm{drew}\:\mathrm{this};\:\mathrm{I}\:\mathrm{took}\:\mathrm{screenshot}\:\mathrm{of} \\ $$$$\mathrm{this}\:\mathrm{figure}\:\mathrm{from}\:\mathrm{a}\:\mathrm{typed}\:\mathrm{question}\:\mathrm{paper} \\ $$$$\mathrm{PDF}. \\ $$
Answered by ajfour last updated on 02/Sep/17
both land on the unshaded part.  points on the radius ahead of P  had lesser velocities along x than  P had.    & points on the radius ahead of  Q had greater velocities along x  than Q had.
$${both}\:{land}\:{on}\:{the}\:{unshaded}\:{part}. \\ $$$${points}\:{on}\:{the}\:{radius}\:{ahead}\:{of}\:{P} \\ $$$${had}\:{lesser}\:{velocities}\:{along}\:{x}\:{than} \\ $$$${P}\:{had}. \\ $$$$\:\:\&\:{points}\:{on}\:{the}\:{radius}\:{ahead}\:{of} \\ $$$${Q}\:{had}\:{greater}\:{velocities}\:{along}\:{x} \\ $$$${than}\:{Q}\:{had}. \\ $$
Commented by Tinkutara last updated on 02/Sep/17
Thank you very much Sir!
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{Sir}! \\ $$

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