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Question Number 5546 by FilupSmith last updated on 19/May/16
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?
$$\mathrm{I}\:\mathrm{have}\:\mathrm{two}\:\mathrm{circles}\:\mathrm{side}\:\mathrm{by}\:\mathrm{side}. \\ $$$$\mathrm{Circle}\:\mathrm{1}\:\mathrm{has}\:\mathrm{radius}\:{r}. \\ $$$$\mathrm{Circle}\:\mathrm{2}\:\mathrm{has}\:\mathrm{radius}\:\frac{\mathrm{1}}{\mathrm{3}}{r}. \\ $$$$ \\ $$$$\mathrm{If}\:\mathrm{I}\:\mathrm{roll}\:\mathrm{circle}\:\mathrm{2}\:\mathrm{around}\:\mathrm{circle}\:\mathrm{1}\:\mathrm{until} \\ $$$$\mathrm{it}\:\mathrm{reaches}\:\mathrm{the}\:\mathrm{begining}, \\ $$$$\mathrm{how}\:\mathrm{many}\:\mathrm{times}\:\mathrm{will}\:\mathrm{it}\:\mathrm{roll}? \\ $$
Commented by FilupSmith last updated on 19/May/16
Correct if wrong but i think i have my answer. Mapping a point on the circle, it produces a sine wave (if the circle is stationary) the length of a sine wave y=sin(x) is equal to π Length=∫_a ^( b) (√(1+((dy/dx))^2 ))dx=∫_0 ^( 2π) (√(1+cos x))dx=π from what i have gathered this means id that by the time it reaches 2π, the curve length is π. similarly, once the spining coin makes it half way around the other coin (π distance), it rotates 2π=360^o sorry for bad or incorrect solution
$$\mathrm{Correct}\:\mathrm{if}\:\mathrm{wrong}\:\mathrm{but}\:\mathrm{i}\:\mathrm{think}\:\mathrm{i}\:\mathrm{have} \\ $$$$\mathrm{my}\:\mathrm{answer}. \\ $$$$ \\ $$$$\mathrm{Mapping}\:\mathrm{a}\:\mathrm{point}\:\mathrm{on}\:\mathrm{the}\:\mathrm{circle},\:\mathrm{it}\:\mathrm{produces} \\ $$$$\mathrm{a}\:\mathrm{sine}\:\mathrm{wave}\:\left(\mathrm{if}\:\mathrm{the}\:\mathrm{circle}\:\mathrm{is}\:\mathrm{stationary}\right) \\ $$$$ \\ $$$$\mathrm{the}\:\mathrm{length}\:\mathrm{of}\:\mathrm{a}\:\mathrm{sine}\:\mathrm{wave}\:{y}=\mathrm{sin}\left({x}\right) \\ $$$$\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\:\pi \\ $$$${Length}=\int_{{a}} ^{\:{b}} \sqrt{\mathrm{1}+\left(\frac{{dy}}{{dx}}\right)^{\mathrm{2}} }{dx}=\int_{\mathrm{0}} ^{\:\mathrm{2}\pi} \sqrt{\mathrm{1}+\mathrm{cos}\:{x}}{dx}=\pi \\ $$$$ \\ $$$$\mathrm{from}\:\mathrm{what}\:\mathrm{i}\:\mathrm{have}\:\mathrm{gathered}\:\mathrm{this}\:\mathrm{means}\:\mathrm{id} \\ $$$$\mathrm{that}\:\mathrm{by}\:\mathrm{the}\:\mathrm{time}\:\mathrm{it}\:\mathrm{reaches}\:\mathrm{2}\pi,\:{the}\:{curve} \\ $$$${length}\:{is}\:\pi. \\ $$$$ \\ $$$$\mathrm{similarly},\:\mathrm{once}\:\mathrm{the}\:\mathrm{spining}\:\mathrm{coin}\:\mathrm{makes}\:\mathrm{it} \\ $$$$\mathrm{half}\:\mathrm{way}\:\mathrm{around}\:\mathrm{the}\:\mathrm{other}\:\mathrm{coin}\:\left(\pi\:{distance}\right), \\ $$$$\mathrm{it}\:\mathrm{rotates}\:\mathrm{2}\pi=\mathrm{360}^{\mathrm{o}} \\ $$$$ \\ $$$${sorry}\:{for}\:{bad}\:{or}\:{incorrect}\:{solution} \\ $$
Commented by Yozzii last updated on 20/May/16
The integral expression for the arc length of the sine curve cannot be expressed in terms of elementary functions. This is because no such result can be the antiderivative of (√(1+cos^2 x)). For the closed interval [0,2π] the arc length ∫_0 ^(2π) (√(1+cos^2 x))dx≈7.6≠π.
$${The}\:{integral}\:{expression}\:{for}\:{the}\:{arc} \\ $$$${length}\:{of}\:{the}\:{sine}\:{curve}\:{cannot}\:{be} \\ $$$${expressed}\:{in}\:{terms}\:{of}\:{elementary}\:{functions}. \\ $$$${This}\:{is}\:{because}\:{no}\:{such}\:{result}\:{can}\:{be} \\ $$$$\:{the}\:{antiderivative}\:{of}\:\sqrt{\mathrm{1}+{cos}^{\mathrm{2}} {x}}. \\ $$$${For}\:{the}\:{closed}\:{interval}\:\left[\mathrm{0},\mathrm{2}\pi\right]\:{the}\:{arc} \\ $$$${length}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \sqrt{\mathrm{1}+{cos}^{\mathrm{2}} {x}}{dx}\approx\mathrm{7}.\mathrm{6}\neq\pi. \\ $$
Commented by FilupSmith last updated on 19/May/16
Commented by Yozzii last updated on 19/May/16
2πr×(1/(2πr/3))=3 times
$$\mathrm{2}\pi{r}×\frac{\mathrm{1}}{\mathrm{2}\pi{r}/\mathrm{3}}=\mathrm{3}\:{times} \\ $$$$ \\ $$
Commented by Yozzii last updated on 19/May/16
The mobile point,say P, on C2 will move covering a distance of 2πr which is the circumference of C1. But, in one revolution of C2 about its centre P covers a distance of only ((2πr)/3). So, for the total distance 2πr to be covered by P, C2 will make ((2πr)/(2πr/3))=3 complete revolutions to return to B. Generally, for circles C1 and C2 having radii r_1 and r_2 respectively, if C1 is stationary and C2 is made to roll along the circumference of C1, then C2 makes ((2πr_1 )/(2πr_2 ))=(r_1 /r_2 ) revolutions about its centre.
$${The}\:{mobile}\:{point},{say}\:{P},\:{on}\:{C}\mathrm{2}\:{will}\:{move} \\ $$$${covering}\:{a}\:{distance}\:{of}\:\mathrm{2}\pi{r}\:{which}\:{is} \\ $$$${the}\:{circumference}\:{of}\:{C}\mathrm{1}.\:{But},\:{in}\:{one} \\ $$$${revolution}\:{of}\:{C}\mathrm{2}\:{about}\:{its}\:{centre}\: \\ $$$${P}\:\:{covers}\:{a}\:{distance}\:{of}\:{only}\:\frac{\mathrm{2}\pi{r}}{\mathrm{3}}. \\ $$$${So},\:{for}\:{the}\:{total}\:{distance}\:\mathrm{2}\pi{r}\:{to}\:{be} \\ $$$${covered}\:{by}\:{P},\:{C}\mathrm{2}\:{will}\:{make}\:\frac{\mathrm{2}\pi{r}}{\mathrm{2}\pi{r}/\mathrm{3}}=\mathrm{3} \\ $$$${complete}\:{revolutions}\:{to}\:{return}\:{to}\:{B}. \\ $$$${Generally},\:{for}\:{circles}\:{C}\mathrm{1}\:{and}\:{C}\mathrm{2}\:{having} \\ $$$${radii}\:{r}_{\mathrm{1}} \:{and}\:{r}_{\mathrm{2}} \:{respectively},\:{if}\:{C}\mathrm{1}\:{is} \\ $$$${stationary}\:{and}\:{C}\mathrm{2}\:{is}\:{made}\:{to}\:{roll} \\ $$$${along}\:{the}\:{circumference}\:{of}\:{C}\mathrm{1},\:{then} \\ $$$${C}\mathrm{2}\:\:{makes}\:\frac{\mathrm{2}\pi{r}_{\mathrm{1}} }{\mathrm{2}\pi{r}_{\mathrm{2}} }=\frac{{r}_{\mathrm{1}} }{{r}_{\mathrm{2}} }\:{revolutions}\:{about} \\ $$$${its}\:{centre}. \\ $$
Commented by FilupSmith last updated on 19/May/16
I thought so, too. What I found Is that if i have two identical coins, coin 1 will rotate TWICE as if does a full cycle. I dont know why
$$\mathrm{I}\:\mathrm{thought}\:\mathrm{so},\:\mathrm{too}.\:\mathrm{What}\:\mathrm{I}\:\mathrm{found}\:\mathrm{Is}\:\mathrm{that} \\ $$$$\mathrm{if}\:\mathrm{i}\:\mathrm{have}\:\mathrm{two}\:\mathrm{identical}\:\mathrm{coins},\:\mathrm{coin}\:\mathrm{1}\:\mathrm{will} \\ $$$$\mathrm{rotate}\:\mathrm{TWICE}\:\mathrm{as}\:\mathrm{if}\:\mathrm{does}\:\mathrm{a}\:\mathrm{full}\:\mathrm{cycle}. \\ $$$$\mathrm{I}\:\mathrm{dont}\:\mathrm{know}\:\mathrm{why} \\ $$
Commented by Yozzii last updated on 19/May/16
So the mobile coin revolved through 720° while both coins are identical?
$${So}\:{the}\:{mobile}\:{coin}\:{revolved}\:{through} \\ $$$$\mathrm{720}°\:{while}\:{both}\:{coins}\:{are}\:{identical}? \\ $$
Commented by FilupSmith last updated on 19/May/16
correct! try it yourself!
$$\mathrm{correct}!\:\mathrm{try}\:\mathrm{it}\:\mathrm{yourself}! \\ $$
Commented by FilupSmith last updated on 19/May/16
I can′t understand why this happenes!
$$\mathrm{I}\:\mathrm{can}'\mathrm{t}\:\mathrm{understand}\:\mathrm{why}\:\mathrm{this}\:\mathrm{happenes}! \\ $$
Commented by FilupSmith last updated on 20/May/16
Oh, i see, i reaslised my mistake. So is there a way to determine its length?
$$\mathrm{Oh},\:\mathrm{i}\:\mathrm{see},\:\mathrm{i}\:\mathrm{reaslised}\:\mathrm{my}\:\mathrm{mistake}. \\ $$$$\mathrm{So}\:\mathrm{is}\:\mathrm{there}\:\mathrm{a}\:\mathrm{way}\:\mathrm{to}\:\mathrm{determine}\:\mathrm{its}\:\mathrm{length}? \\ $$
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
I am yet to reach such a level of Calculus and Analysis to explain the elliptic integral. I also cannot say that there isn′t a way to find the exact arc length of the sine curve in terms of elementary functions.
$${I}\:{am}\:{yet}\:{to}\:{reach}\:{such}\:{a}\:{level}\:{of} \\ $$$${Calculus}\:{and}\:{Analysis}\:{to}\:{explain} \\ $$$${the}\:{elliptic}\:{integral}.\:{I}\:{also}\:{cannot} \\ $$$${say}\:{that}\:{there}\:{isn}'{t}\:{a}\:{way}\:{to} \\ $$$${find}\:{the}\:{exact}\:{arc}\:{length}\:{of}\:{the}\:{sine} \\ $$$${curve}\:{in}\:{terms}\:{of}\:{elementary}\:{functions}. \\ $$$$ \\ $$

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