I-have-two-circles-side-by-side-Circle-1-has-radius-r-Circle-2-has-radius-1-3-r-If-I-roll-circle-2-around-circle-1-until-it-reaches-the-begining-how-many-times-will-it-roll-

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Question Number 5546 by FilupSmith last updated on 19/May/16

$$\mathrm{I}\mathrm{have}\mathrm{two}\mathrm{circles}\mathrm{side}\mathrm{by}\mathrm{side}.$$$$\mathrm{Circle}1\mathrm{has}\mathrm{radius}r.$$$$\mathrm{Circle}2\mathrm{has}\mathrm{radius}\frac{1}{3}r.$$$$$$$$\mathrm{If}\mathrm{I}\mathrm{roll}\mathrm{circle}2\mathrm{around}\mathrm{circle}1\mathrm{until}$$$$\mathrm{it}\mathrm{reaches}\mathrm{the}\mathrm{begining},$$$$\mathrm{how}\mathrm{many}\mathrm{times}\mathrm{will}\mathrm{it}\mathrm{roll}?$$

Commented by FilupSmith last updated on 19/May/16

$$\mathrm{Correct}\mathrm{if}\mathrm{wrong}\mathrm{but}\mathrm{i}\mathrm{think}\mathrm{i}\mathrm{have}$$$$\mathrm{my}\mathrm{answer}.$$$$$$$$\mathrm{Mapping}\mathrm{a}\mathrm{point}\mathrm{on}\mathrm{the}\mathrm{circle},\mathrm{it}\mathrm{produces}$$$$\mathrm{a}\mathrm{sine}\mathrm{wave}\left(\mathrm{if}\mathrm{the}\mathrm{circle}\mathrm{is}\mathrm{stationary}\right)$$$$$$$$\mathrm{the}\mathrm{length}\mathrm{of}\mathrm{a}\mathrm{sine}\mathrm{wave}y=\sin \left(x\right)$$$$\mathrm{is}\mathrm{equal}\mathrm{to}\pi$$$$Length={\int }_a^b\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}dx={\int }_0^{2\pi }\sqrt{1+\cos x}dx=\pi$$$$$$$$\mathrm{from}\mathrm{what}\mathrm{i}\mathrm{have}\mathrm{gathered}\mathrm{this}\mathrm{means}\mathrm{id}$$$$\mathrm{that}\mathrm{by}\mathrm{the}\mathrm{time}\mathrm{it}\mathrm{reaches}2\pi ,thecurve$$$$lengthis\pi .$$$$$$$$\mathrm{similarly},\mathrm{once}\mathrm{the}\mathrm{spining}\mathrm{coin}\mathrm{makes}\mathrm{it}$$$$\mathrm{half}\mathrm{way}\mathrm{around}\mathrm{the}\mathrm{other}\mathrm{coin}\left(\pi distance\right),$$$$\mathrm{it}\mathrm{rotates}2\pi ={360}^{\mathrm{o}}$$$$$$$$sorryforbadorincorrectsolution$$

Commented by Yozzii last updated on 20/May/16

$$Theintegralexpressionforthearc$$$$lengthofthesinecurvecannotbe$$$$expressedintermsofelementaryfunctions.$$$$Thisisbecausenosuchresultcanbe$$$$theantiderivativeof\sqrt{1+{cos}^{2}x}.$$$$Fortheclosedinterval[0,2\pi ]thearc$$$$length{\int }_0^{2\pi }\sqrt{1+{cos}^{2}x}dx\approx 7.6\ne \pi .$$

Commented by FilupSmith last updated on 19/May/16

Commented by Yozzii last updated on 19/May/16

$$2\pi r\times \frac{1}{2\pi r/3}=3times$$$$$$

Commented by Yozzii last updated on 19/May/16

$$Themobilepoint,sayP,onC2willmove$$$$coveringadistanceof2\pi rwhichis$$$$thecircumferenceofC1.But,inone$$$$revolutionofC2aboutitscentre$$$$Pcoversadistanceofonly\frac{2\pi r}{3}.$$$$So,forthetotaldistance2\pi rtobe$$$$coveredbyP,C2willmake\frac{2\pi r}{2\pi r/3}=3$$$$completerevolutionstoreturntoB.$$$$Generally,forcirclesC1andC2having$$$$radii{r}_{1}and{r}_{2}respectively,ifC1is$$$$stationaryandC2ismadetoroll$$$$alongthecircumferenceofC1,then$$$$C2makes\frac{2\pi r_1}{2\pi r_2}=\frac{r_1}{r_2}revolutionsabout$$$$itscentre.$$

Commented by FilupSmith last updated on 19/May/16

$$\mathrm{I}\mathrm{thought}\mathrm{so},\mathrm{too}.\mathrm{What}\mathrm{I}\mathrm{found}\mathrm{Is}\mathrm{that}$$$$\mathrm{if}\mathrm{i}\mathrm{have}\mathrm{two}\mathrm{identical}\mathrm{coins},\mathrm{coin}1\mathrm{will}$$$$\mathrm{rotate}\mathrm{TWICE}\mathrm{as}\mathrm{if}\mathrm{does}\mathrm{a}\mathrm{full}\mathrm{cycle}.$$$$\mathrm{I}\mathrm{dont}\mathrm{know}\mathrm{why}$$

Commented by Yozzii last updated on 19/May/16

$$Sothemobilecoinrevolvedthrough$$$$720°whilebothcoinsareidentical?$$

Commented by FilupSmith last updated on 19/May/16

$$\mathrm{correct}!\mathrm{try}\mathrm{it}\mathrm{yourself}!$$

Commented by FilupSmith last updated on 19/May/16

$$\mathrm{I}{\mathrm{can}}^{\prime }\mathrm{t}\mathrm{understand}\mathrm{why}\mathrm{this}\mathrm{happenes}!$$

Commented by FilupSmith last updated on 20/May/16

$$\mathrm{Oh},\mathrm{i}\mathrm{see},\mathrm{i}\mathrm{reaslised}\mathrm{my}\mathrm{mistake}.$$$$\mathrm{So}\mathrm{is}\mathrm{there}\mathrm{a}\mathrm{way}\mathrm{to}\mathrm{determine}\mathrm{its}\mathrm{length}?$$

Commented by Yozzii last updated on 20/May/16

Commented by Yozzii last updated on 20/May/16

Commented by Yozzii last updated on 20/May/16

$$Iamyettoreachsuchalevelof$$$$CalculusandAnalysistoexplain$$$$theellipticintegral.Ialsocannot$$$$saythatthere{isn}^{\prime }tawayto$$$$findtheexactarclengthofthesine$$$$curveintermsofelementaryfunctions.$$$$$$

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