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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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