Two disc are suspended with a ribbon between them as shown in figure. At t $$=$$ $$0$$ the ribbon has $$100$$ complete $$anti-clock$$ twists in it, i.e., the ribbon is twisted by keeping the upper disc fixed and rotating the lower disc $$anti-clock$$ when viewed from above. At t $$=$$ $$0$$ the upper disc starts to rotate at ω$$u=2rpm$$ anticlockwise, and the lower disc, starting from rest, begins to rotate with a clockwise angular acceleration of a magnitude of $$1$$ $$rpm/s2$$.At what time tf does the ribbon becomes momentarily untwisted?

Question

Two disc are suspended with a ribbon between them as shown in figure. At t $$=$$ $$0$$ the ribbon has $$100$$ complete $$anti-clock$$ twists in it, i.e., the ribbon is twisted by keeping the upper disc fixed and rotating the lower disc $$anti-clock$$ when viewed from above. At t $$=$$ $$0$$ the upper disc starts to rotate at  ω$$u=2rpm$$ anticlockwise, and the lower disc, starting from rest, begins to rotate with a clockwise angular acceleration of a magnitude of $$1$$ $$rpm/s2$$.At what time tf does the ribbon becomes momentarily untwisted?

Infinity Learn · Accepted Answer

Both disc rotate in such a way that ribbon is untwisted. i.e., the upper disc will rotates at constant angular speed (approximately) and lower disc will rotates at constant angular acceleration (approximately).          Angles rotated by upper disc $$=$$ θu $$=$$ ωutf          Lower disc $$=$$ θL $$=$$ ωLtf $$+12aL$$ $$tf2i$$.e., relative angular displacement ; (100)2$$π $$=$$ $$22$$$$π$$$$60$$ tf $$+$$ $$12$$ $$(2$$$$π$$$$60) $$tf2tf$$ $$=$$ $$107.56$$ $$sec$$.Angles rotated by upper disc $$=$$  θ$$u=260$$×$$107.56$$θ$$u=3.585rev$$ Angle rotated by lower disc $$=$$  θ$$L=12$$α$$Ltf2=$$ $$12160$$×$$(107.56)2$$ $$=$$ $$96.407Relative$$ number of turns $$=$$ $$3.585$$ $$+$$ $$96.407$$ $$=$$ $$99.99$$≈$$100rev$$

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