IB HL Maths - Maths Exploration - Cycloid.docx

St. Joseph Institution International

Pham Minh  Tue  T ue

 Table  Table of conten

[MODELING CYCLOID]

A

An investigation about the properties of a cycloid

 Table  Table of content................................. content..................................................... ....................................... ....................................... ................................ .............. .. 1 Rationale................................ Rationale.................................................... ........................................ ............................................................ ............................................ ....  Introduction................................... Introduction............... ....................................... ....................................... .......................................................... ...................................... ! "artesian e#uation for cycloid............................................... cycloid................................................................... .................................... .................. $ %ori&ontal movement of the 'hole circle................................................................$ Rotational movement of point A relative to point (................................................)  The "artesian e#uation for cycloid......................................... cycloid.......................................................................... ................................. ) Re*ection on the process of deriving the e#uation.................................................+ Area under a cycloid arc................................. arc..................................................... ........................................ ..................................... ................. ,  The -rachistochrone curve and the Tautochrone Tautochrone curve.................................... curve.............................................. ..........   Transforming  Transforming the current "artesian e#uation of the cycloid.................................... Initial conditions and de/nitions................................................... de/nitions............................................................................ .........................10 10 Sho'ing that the cycloid has h as the Tautochrone Tautochrone property........................................11 property........................................11 Application of the cycloid.........................................................................................1 2sing the Tautochronous Tautochronous property to improve mechanical cloc3s....................... ..1  Testing  Testing the tautochronous property using computer simulation......... ................. ........... .......1) .1) -ibliography.............................................................................................................1,

Page 4 

Rationale As a child5 loo3ing at people riding bicycles5 I often 'ondered 6ust in 'hat 'ay particular points of the 'heel move. "ertainly5 by that time5 I had no idea ho' I could ever ans'er this #uestion 'ith my limited mathematical 3no'ledge. A fe' years later5 after learning about the basics of speed and time5 I reali&ed that the center of the 'heel only moves in a straight5 hori&ontal line 'ith the speed e#ual to that of the bicycle 7 a recognition that directly coincide 'ith my constant observations throughout the years. %o'ever5 that recognition 6ust ans'ers a little5 simple part of my initial #uestion 7 'hat about the movement of a point on the rim of the 'heel8 As time passed by5 this #uestion seemingly disappeared from my mind as I seldom thought about it5 and I 'ould probably completely forgot about them if I had not found this little story 6ust some months ago5 'hich captivates me almost immediately. The story goes as follo'ed9 In June 1))5 Johann -ernoulli :1))+ 7 1+$,; proposed a problem that later is 3no'n as ive mathematicians responded to the problem5 including Isaac ?e'ton5 Ja3ob -ernoulli :Johann@s brother;5 ottfried Beibni&5 Chrenfried Dalther Eon Tschirnhaus and uillaume de lF%Gpital. It is said that ?e'ton found the problem in his mail 'hen he arrived home from the mint at $ p.m.5 and stayed up all night to solve it and mailed the solution by the neHt post. This sho's ho' good ?e'ton is5 because for the same problem5 it too3 Johann himself  'ee3s to solve5 only to come up 'ith a marred solution because he did not ta3e into consideration all of the constants :.T.Dhiteside;. The solution of the problem is the cycloid :The "ycloid5 01;. (f course5 the story is so interesting to me because it involves a lot of famous scientists and mathematicians of that time. It really ma3es me 'onder 6ust ho' important and challenging this problem is so that it attracts such great people to solve it independently from each other. Thus5 I 'as very curious about this and /nally decided to investigate more about this speci/c curve 7 the cycloid. To my surprise5 the shape of this curve is eHactly the one #uestioned by me earlier 7 the path of a point on the circumference of a circle as the circle rolls along a straight line :The "ycloid5 01;.

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Introduction As de/ned above5 the cycloid is the path of a point on the circumference of a circle as the circle rolls along a straight line :The "ycloid5 01;. The main aim of this eHploration is to investigate the properties of the cycloid as such5 I 'ill not only focus on deriving the "artesian e#uation for a cycloid5 but also consider other characteristics such as the area under the curve. In order to do this5 I 'ill plot the vertical and hori&ontal displacements on the yKaHis and the HKaHis of a graph respectively to see their relationship as 'ell as eHploring other properties based on the analysis of the "artesian grid. Moreover5 I 'ould attempt to solve famous problems involving the cycloid such as the above -rachistochrone problem 'henever appropriate. At the end of this eHploration5 I have succeeded in deriving the "artesian e#uation for the "ycloid and calculating the area under a cycloid arc. Additionally5 I have successfully proved that the cycloid is indeed a solution to the Tautochrone problem :the problem of  /nding a curve do'n 'hich a particle placed any'here 'ill fall to the bottom in the same amount of time5 under nothing but its o'n 'eight :Deisstein;;5 but I could not prove that it is the only solution because it is beyond the I- syllabus. ?onetheless5 in my eHploration5 I have also successfully tested the tautochronous property of the cycloid by using a computer simulation. Moreover5 I have eHplored the application of this particular property by eHploring ho' this property can be used to improve the accuracy of a mechanical cloc3. %o'ever5 regarding the -rachistochrone problem5 I could not prove that the cycloid is the solution to the -rachistochrone problem since it is beyond the content of the Isyllabus.  To pro6ect the shape of the cycloid5 I decided to ma3e some initial conditions9 The radius of the rolling circle is r > 0 5 the velocity of the circle rolling along the hori&ontal line is v O > 0 . The circle 'ill start rolling at such a position that the center ( of the circle is at :05r; and the point A on the circumference of the circle is at A o L :050;. The angle at 'hich point A is from its initial position A o 'ith regards to the center ( is θ  radians5 for 'hich :in other 'ords5 I 'ill only investigate the /rst arc of the cycloid;. The 0 < θ < 2 π  movement of the circle is from left to right for convenience purpose. Important factors aecting the movement of point A are the hori&ontal movement of every point on the circle as 'ell as the rotational movement of the point A as the circle rolls. I 'ould pro6ect the shape of the cycloid in my eHploration by using the computer program Autograph !.!

Page 4 $

Object 1: A cycloid arc generated by a rolling circle :Surendran5 The "ycloid;

"artesian e#uation for cycloid  To derive a "artesian e#uation for the cycloid5 I decided to split the process into  sections9 investigating the hori&ontal movement of the 'hole circle and investigating the rotational movement of point A. After that5 I 'ould combine the  movements in a vectorK li3e manner to come up 'ith the /nal e#uation of the cycloid.

Horiontal !o"e!ent o# t$e %$ole circle

Object &: Horiontal !o"e!ent o# a cycloid

In the graph above5 'e are only concerned 'ith the movement of the point ( as the circle rolls along the hori&ontal road. As abovementioned5 the 'hole circle is rolling at the velocity vo. In the follo'ing section5 I 'ill derive the hori&ontal and vertical displacements of point ( as a function of the time t. The hori&ontal displacement and vertical displacement are labeled as  x O ( t )  and  y O ( t )  respectively. Since velocity is the derivative of displacement5 'e have9 d  x ( t )= vO dt  O

Page 4 

d x O ( t )= v O dt 

∫v

 x O=

O

dt =v O ×t ( 1 )

De can see that as time t passes5 the circle moves a distant of %ence5 'e have the graph belo'9

v O ×t    hori&ontally.

Object ': ($e $oriontal and rotational !o"e!ent) o# *oint A

As time t passes5 point A has rotated the angle θ  relative to point (. As the circle rolls a distant of v o ×t  5 point A also rotates a total distant of arc length A 1At. Moreover5 'e also have9 arc length A 1 A t =r ×θ

 Thus5 'e have9 v O ×t = r × θ

C#uation :1; no' becomes9  x O=r ×θ ( 2)

>or the vertical displacement of point (5 'e can clearly see that no matter the position of  the circle5 point ( is still r units above the HKaHis. Thus5 the vertical displacement of point ( after time t   is a constant r 5 'here r  is the radius of the circle. De have9  y O= r ( 3 )

Page 4 )

 Therefore5 'e come to a conclusion that5 after time displacements are given by9

t  5 the hori&ontal and vertical

 x O=r ×θ ( 2)  y O= r ( 3 )

Page 4 +

+otational !o"e!ent o# *oint A relati"e to *oint O

Object ': ($e rotational !o"e!ent o# *oint A relati"e to *oint O

As seen in the graph above5 I can divide the displacement of A relative to ( into t'o displacements9 the hori&ontal displacement  x AO  and the vertical displacement  y AO .  These  displacements can be eHpressed by9  x AO=−r × sin θ ( 4 )  y AO=−r × cos θ ( 5 )

($e Carte)ian e,-ation #or cycloid De can combine the hori&ontal movement of point ( and the rotational movement of  point A relative to point ( to result in the "artesian e#uation for the cycloid in the same manner as adding vector 'hich5 in this case5 the e#uation 'ill be9  x A= x O + x AO

( 2 ) +( 4 ) → x A= rθ− r sin θ  y A = y O + y AO

( 3 ) +( 5 ) → y A =r −r cos θ In conclusion5 the "artesian e#uation for cycloid is9  x =rθ −r sin θ (6 )  y =r − r cos θ ( 7 )

Page 4 ,

+e.ection on t$e *roce)) o# deri"ing t$e e,-ation As abovementioned5 one of my initial conditions is that the circle rolls at a constant speed of v O . This seemingly becomes a problem because the circle does not necessary move at a constant speed. %o'ever5 if 'e ta3e a loo3 at the /nal "artesian e#uation of the cycloid :e#uations :); and :+;;5 'e can see that there are no v O   and t    presented.  This suggests that these  variables are not relevant in this case5 and thus 'e don@t need this condition of the speed of the circle. In the follo'ing section5 I 'ill revie' on the process of deriving the e#uation to see if this is indeed the case. Indeed5 the only time that I used movement of point (9

∫v

 x O=

O

v O   and

t   is 'hen I 'or3ed out the hori&ontal

dt =v O ×t ( 1 )

%o'ever5 I soon reali&ed that v O ×t = r × θ   and thus  x O=r ×θ ( 2) . At any point after v O   and t  are not important anymore to the process. this5 the t'o variables  Therefore5 instead of deriving the hori&ontal displacement  x A and the vertical displacement  y A  in term of time5 the /nal e#uations for these t'o displacements are derived in term of θ 5 'hich is independent of both time t   and speed v O . %o'ever5 these  variables are still helpful to the process in the sense that they help us to visuali&e the movement better5 even though they do not contribute anything to the /nal result.

Page 4 

Area -nder a cycloid arc Previously5 'e have these t'o e#uations9  x A=rθ −r sin θ ( 6 )  y A =r −r cos θ ( 7 ) →

d x =r −r cos θ dθ  A

→ d x A =( r −r cos θ ) dθ θ= 0  and

 The area under the curve of a cycloid arc :limited by

θ= 2 π   is9

2  π 

∫

 Area=  y A d x A 0

r 2 π 

(¿ ¿ 2−2 r

2

cos θ

+r

2

cos

2

θ ) dθ=r

2

∫ (1 −2cos θ +cos θ ) dθ 2

0 2 π 

2 π 

0

0

∫ ( r − r cos θ ) × ( r−r cos θ ) dθ=∫ ¿

→ Area=

(

→ Area= θ−2 sin θ +

→ Area=r

2

[(

3 2

=3 π r

∴ Area

sin 2 θ 4

+

θ 2

) (

× 2 π −2sin2 π +

2 π 

=

0

3

θ− 2sin θ +

2

)(

sin 4 π  4

−

3 2

sin 2 θ 4

)

× 0−2sin 0 +

2 π 

0

sin 0 4

)] [( =r

2

− 2× 0+

3 π 

0 4

)

0

]

−(−2 × 0 + ) 4

2

Regarding the area under a cycloid arc5 there is a story involving another great scientist as follo'ed9 Galileo Galilei (interestingly enough, also the one who named the curve cycloid in 1599 (Robertson)) once attempted to fnd the area by comparing its area to that o the generating circle !ter ailing to fnd a mathematical method he resorted to weighing pieces o metal cut into the shape o the cycloid "e ound that the ratio o the weights was appro#imately $ to 1 but decided that it was not e#actly $ because he guessed that the ratio was not rational (Robertson)

 This story immediately caught my attention5 as I found it very astonishing that not only ?e'ton and the -ernoulli brothers 'ere involved in the research of this special curve5 but alileo K
that started the investigation on this ama&ing curve. It is 'orth noting that this attempt 'as made before Isaac ?e'ton and ottfried Beibni& discovered calculus in the 1+ th century :-urton;. Surprisingly enough5 this empirical method that 'as deemed 'rong by alileo actually yielded correct ans'er as 'e can see above9 the area under a cycloid curve is 3 π r 2 5 eHactly ! times the area of the circle generating the cycloid :area of the circle is

πr

2

;

 The -rachistochrone curve and the  Tautochrone curve  The cycloid is also signi/cant because it is the solution to the famous -rachistochrone problem :the one ?e'ton solved in one day; and the Tautochrone problem. The -rachostochrone problem is the problem of /nding a curve along 'hich a particle can move from the upper point to the lo'er point :on the same vertical plane; in the shortest amount of time5 under nothing but its o'n 'eight. :The "ycloid5 01;. The proof of this problem is beyond the syllabus of I-5 so I 'ill not attempt to prove it. The Tautochrone problem is the problem of /nding a curve do'n 'hich a particle placed any'here 'ill fall to the bottom in the same amount of time5 under nothing but its o'n 'eight :Deisstein;. I 'ill not try to prove that the cycloid is the only solution to this problem because the proof  is also beyond the I- syllabus. Instead5 in the follo'ing section5 I 'ill sho' that the cycloid does satisfy the condition of the problem and thus5 is a solution to the  Tautochrone problem.

(ran)#or!ing t$e c-rrent Carte)ian e,-ation o# t$e cycloid As the Tautochrone problem deals 'ith gravity5 it is inappropriate for the current e#uation of the cycloid because the curve is facing do'n'ard. Thus5 I 'ill transform the curve to a more suitable one by re*ecting it on the HKaHis and then translating it up'ard by 2 r units. It is 'orth noting that I only modify the yKvalue of the curve5 so the HKvalue of the curve is left unchanged.

///////

///////

Page 4 11

Object : ($e tran)#or!ation o# t$e cycloid

 The current yKvalue of the cycloid is given by  y =r −r cos θ −−−( 7 ) . After re*ecting the curve on the HKaHis5 the yKvalue becomes  y =r cos θ −r . After translating the curve by  y =r cos θ −r + 2 r = r + r cos θ . Thus5 the e#uation 2 r  units up'ard5 the yKvalue becomes of the ne' cycloid is9  x =rθ −r sin θ (6 )  y =r + r cos θ ( 8 )

Initial condition) and de2nition) -efore proceeding5 I 'ill need to ma3e some initial conditions and de/nitions. I 'ill let the point along the cycloid curve that the particle starts its movement be  M 0=( x 0 , y 0) . This  M 0   point can be any point bet'een the highest point and the lo'est point of the cycloid and it is a /Hed point. As the particle starts moving from the point  M 0 5 I 'ill label the initial time as t 0=0  and the initial angle as θ0 . These  variables are also /Hed variables li3e the point  M 0 . The particle 'ill move from  M 0  to an intermediate point  M =( x , y )   bet'een  M 0 and the lo'est point of the cycloid5 'here the coordinates of  M   :in other 'ords5  x   and  y ; are functions of θ 9 these values of 5  x   and  y  are the same variables as the one presented in the e#uations of the cycloid above :e#uation :); and :,;;. The ranges of  x  and  y  are 0 ≤ x ≤ 2 πr  :from e#uation :);; and 0 ≤ y ≤ 2 r   :from e#uation :,;;. The velocity of the particle at point  M   is labeled as v  and is a function of θ  'here v > 0 . The time for the particle to move from point  M 0   to point  M    is labeled as t    and is a function of θ . The length of the curve  M 0 M > 0  is labeled as s  and is also a function of θ . De can derive from the information already presented that the angle at the lo'est point of the cycloid is π   :the middle point bet'een the  limiting angles 0  and 2 π  ; and the yKcoordinate of this point is 0 . Also5 at point  M 0 5 the particle 6ust starts moving5 so the velocity of the particle at  M 0  is v 0 =0 .  The time for the particle to move from point

 M 0  to the lo'est point of the cycloid is9

t lowest 

T =

∫ dt  0

%&ote' the variables v , t   and s  used in this section are completely dierent rom the one using above in the *artesian e+uation or cycloid section

Page 4 1

Object 3: Gra*$ )$o%ing t$e !o"e!ent o# t$e *article along t$e cycloid c-r"e

4$o%ing t$at t$e cycloid $a) t$e (a-toc$rone *ro*erty >rom earlier5 'e have9  x =rθ −r sin θ (6 )  y =r + r cos θ ( 8 )

-y ta3ing the derivatives5 'e have9 dx =r − r cos θ=r ( 1−cos θ ) dθ → dx =r ( 1− cos θ ) dθ dy =−r sin θ dθ → dy =−r sin θ dθ

 The curve length ds =√ ( dx )

2

s  is related to the  variables

+ ( dy ) =√ r ( 1 −cos θ ) ( dθ ) +r 2

2

2

2

2

sin

2

θ ( dθ )

 x  and

 y  by9

2

ds =r √ 1−2 cos θ + cos θ + sin θ dθ =r √ 2−2 cos θ dθ ( 9) 2

2

Page 4 1!

 To proceed5 'e have to ta3e into consideration the conservation of energy at all time of  the particle i.e. the loss in gravitational potential energy : GPE =mgy ; of the particle by 1

moving from point  M 0  to point  M   e#uals to the gain in 3inetic energy : KE= m v

2

2

; of the particle. De have9  KE =−GPE 1

→ mv

2

2 1

→ m (v

1

2

0

2

2

2

→v

2

− m v =−( mgy− mg y ) −v ) =mg (  y − y ) 2

0

=2 g ( y − y )   0

→ v=

0

0

:since

v 0 =0 ;

ds = 2 g ( y 0− y ) dt  √ 

Applying e#uation :,; into the above e#uation5 'e have9 θ0 r + r cos ¿

¿

θ r + r cos ¿

−¿ ¿ 2g¿

ds =√ ¿ dt 

θ

−cos ¿ ¿ 2 gr ¿ √ ¿ ds → dt = ¿

cos θ0

Substitute e#uation :; into the above e#uation5 'e have9

Page 4 1$

θ

− cos ¿ ¿ ¿ 2 gr ¿ √ ¿ r √ 2 −2cos θ dθ dt = ¿ cos θ0

At *oint

θ0

θ

At t$e lo%e)t *oint

t lowest 

0

t 

 M 0

π 

Object 5: 6ariable) con"erting table

 The time for the particle to move from point

 M 0  to the lo'est point is9

t lowest 

T =

∫ dt  0

π 

∫

→T =

θ0

√

r  × g

√

√

−cos θ  dθ cos θ −cos θ 1

0

θ  dθ ( 10 ) ∫ √ cos1−θ cos −cos θ π 

r →T =  × g θ

0

0

De have9 cos θ

→ 1 −cos θ

cos θ

() = ( )( ( )− ( )

=1−2 sin

2

θ

2

2sin

2

=2cos

θ

2

2

θ

2

)

11

1 12

Page 4 1

cos θ0

2

=2cos

( )− ( θ0

1 13

2

)

Substitute e#uations :11;5 :1;5 :1!; into e#uation :10;5 'e have9

T =

√

π 

r × g θ

→T =

→T =

→T =

∫

√ √ √

0

√

 

∫ 0

π 

r  × g θ

2

() θ

2

( ( ) )( () ) 2

2cos

π 

r  × g θ

2 sin

∫

√ √

θ0 2

 

2 sin

2cos

2

2

−1 −

2

2cos

()

( )− ( ) 2 cos

2

sin

2

2

cos

() 2

r  × g θ

θ0

2

cos

2

sin

π 

2

cos

θ0 2

dθ

θ

2

dθ

θ

2

θ

2

0

−1

θ

( )− ( ) ( ) ∫ √ ( )− ( ) 0

2

2

θ

2

θ0

θ

dθ

cos

2

dθ =(14 )

θ

2

?o' let9

() ( )

cos

2

!= cos

→

θ

 d! = dθ

θ0 2

1 cos

( ) θ0

×

( ( )) −sin

θ

2

×

 1 2

2

Page 4 1)

d! → = dθ

() ( ) ( ) () θ

−sin 2cos

2

θ0 2

θ0

−2cos → dθ= sin

2

θ

× d! ( 15 )

2

θ0

θ

() ( )

cos

π 

θ

2

!= cos

1

θ0

0

2

Object 7: 6ariable) con"erting table

Substitute e#uation :1; into e#uation :1$;5 'e have9

( ) ( ) √ ∫ ( )− ( ) ( ) √ ( ) =− √ ∫ ( )− ( ) √

r T = × g

sin

0

1

cos

→T 

2

r × g

2

θ

−2 cos

2

θ0

×

cos

2

cos

0

1

cos

2

θ0 2

2

θ

sin

2

θ

θ0 2

×d!

2

θ0 2

cos

d!

2

θ

2

Page 4 1+

→T =2

→T =2

→T =2

√ √ √ √

= π 

∴ T 

r  × g

r  × g

1

∫ 0

1

∫ 0

1

√

cos

cos

2

( )− ( ) ( ) ( ) θ0

2

cos

2

2

θ0

cos

2

1

√ 1 −!

2

1

d!

2

θ

2

θ0 2

d!

√

r r −1  × ( sin ! ) =2  × g g 0

[ sin− ( 1 )−sin− ( 0 ) ]=2 1

1

√ [ ] r π  × −0 g 2

r g

-ecause this is a constant5 the particle ta3es eHactly the same time to move from every point  M 0  along the curve to the lo'est point of the cycloid. Thus5 the cycloid is the solution to the tautochrone problem.

Page 4 1,

Application of the cycloid 8)ing t$e (a-toc$rono-) *ro*erty to i!*ro"e !ec$anical cloc9) Dhat does having the tautochronous property mean8 It means that no matter 'hat the position of the particle is5 it 'ill fall to the bottom in the eHact same amount of time9

√

r . ?o'5 'e 3no' that in a simple pendulum i.e. a bob attached by a thread to a g /Hed point5 the bob 'ill circulate in a circular arc because the distance bet'een the bob and the /Hed point is unchanged because it is the length of the thread. >or a circular pendulum5 the period of oscillation :the time it ta3es for the bob to /nish one complete oscillation; is not perfectly independent of the amplitude :the maHimum hori&ontal displacement from the bob to the /Hed point; i.e. for dierent starting points5 the period of oscillation 'ill be dierent. (f course5 in a frictionless 'orld5 this 'ill not have any eect on the period of oscillation of the cycloid because the amplitude is not aected. %o'ever5 in the real 'orld5 things do not 'or3 that 'ay for there is air friction that 'ill aect the amplitude of the pendulum. This means that as time passes and friction becomes more signi/cant5 the error in time 'ill be larger. π 

So5 the tautochronous property of the cycloid suggests that if 'e can someho' force the bob of the pendulum to move in a cycloidal path rather than a circular path as of no'5 the time period of the oscillation 'ill be 'ay more accurate. This is because as proven5 the time period of oscillation for a cycloidal path is a constant and is perfectly independent of  the amplitude. This means that even if there is air friction5 it 'ill not aect the period of  oscillation as no matter 'here the starting point is5 the period 'ill be a constant. "hristian %uygens :1) 7 1); 'as the /rst person to come up 'ith an ingenious idea of ho' to force the bob to move in a upKsideKdo'n cycloidal path. %e placed  metal plates at the fulcrum of the pendulum5 so that as the bob s'ings up'ard5 the thread 'inds along the plates5 forcing the bob a'ay from its natural circular path :The "ycloid5 01;. %e then discovered that if the shape of the  metal plates is also a cycloid5 then the bob 'ill trace the desired cycloidal path'ay :At3inson;. %uygens@ /rst pendulum cloc3 'as accurate to 1 minute a day after 'or3ing 'ith the best cloc3ma3ers5 he soon made cloc3s that 'ere accurate to 1 second a day :Michael R. Matthe's;.

Page 4 1

Object : A circ-lar *end-l-! ;
Object >: A

(e)ting t$e ta-toc$rono-) *ro*erty -)ing co!*-ter )i!-lation Aside from the above mathematical proof :in the
Object 1?: Circ-lar and Cycloidal *end-l-!) at ? )econd %it$o-t air re)i)tance

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Object 11: Circ-lar and Cycloidal *end-l-!) at &?@5 )econd ;a#ter 3 *eriod)= %it$o-t air re)i)tance

%o'ever5 'hen I changed the setting of the simulation to include in air resistance5 the situation changes considerably. Speci/cally5 the amplitudes of both pendulums decrease as time 'ent on. Additionally5 'hile the time period of the cycloidal pendulum is 3ept constant at $.1 second5 the time period of the circular pendulum decreases considerably. This is demonstrated in the pictures belo'9

Object 1&: Circ-lar and Cycloidal *end-l-!) at ? )econd %it$ air re)i)tance

Object 1': Circ-lar and Cycloidal *end-l-!) at &?@5 )econd %it$ air re)i)tance

(b6ect 1! sho's the  pendulums after 0.) seconds :after  periods of the cycloidal pendulum;. De can see that the bob of the cycloidal pendulum is at its highest point5 even though it cannot reach its initial height any more. This signi/es that that even though the amplitude of the cycloidal pendulum has decreased :it can no longer reach its Page 4 1

previous highest point due to the air resistance;5 its time period still remained constant at $.1 second. Mean'hile5 the simple pendulum@s time period has changed so much after 0.) seconds that 'hen the cycloidal pendulum completes  revolutions :and thus5 its bob is at the highest point in the left side of the e#uilibrium point;5 the bob of the circular pendulum is actually on the right of the e#uilibrium point. Moreover5 by this time5 the circular pendulum 'as currently on its + th period already :'hile the cycloidal pendulum  6ust completed its th  period;5 signifying that the time period of the circular pendulum does in fact decrease 'ith the eHistence of air resistance. This agrees 'ith my statement that a cycloidal pendulum does indeed have the Tautchronous property5 unli3e the simple circular pendulum5 thus signifying that a cycloidal pendulum can in fact be used to improve the accuracy of mechanical cloc3s.  Therefore5 by using this computer simulation5 I have successfully tested the reliability of  my model.

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-ibliography At3inson5 . :n.d.;. "uygens- .endulum . Retrieved ecember +5 01!5 from Theoretical %ighKCnergy Physics9 http9NNthep.housing.rug.nlNsitesNdefaultN/lesNusersNuser1N%uygensOpendulum.pdf  -urton. :n.d.;. /niversity o *aliornia, Riverside 0 epartment o 2athematics . Retrieved ecember $5 01!5 from The development of calculus9 http9NNmath.ucr.eduNresNmath1!Nhistory1.pdf  *ycloid. :n.d.;. Retrieved ecember +5 01!5 from Di3ipedia5 the free encyclopedia9

http9NNen.'i3ipedia.orgN'i3iN"ycloid .T.Dhiteside. ?e'ton the mathematician. In .T.Dhiteside5 &ewton the mathematician :p. 1;. -echler9 "ontemporary ?e'tonian Research. Michael R. Matthe's5 M. P. :n.d.;. 3he Role o 3heory 0 .endulum 2otion, 3ime 2easurement, and the 4hape o the arth . Retrieved ecember +5 01!5 from The Story behind the Science K -ring science and scientists to life9 http9NN'''.storybehindthescience.orgNpdfNearthshape.pdf  .endulum. :n.d.;. Retrieved ecember +5 01!5 from Di3ipedia5 the free encyclopedia9

http9NNen.'i3ipedia.orgN'i3iNPendulum Robertson5 J. J. :n.d.;. *ycloid. Retrieved ecember $5 01!5 from The MacTutor %istory of Mathematics archive9 http9NN'''Khistory.mcs.stKand.ac.u3N"urvesN"ycloid.html Surendran5 . :n.d.;. 3he *ycloid. Retrieved ecember !5 01!5 from 2niversity of Qimbab'e9 http9NNu&'eb.u&.ac.&'NscienceNmathsN&imathsNcycloid.htm 3he *ycloid. :015 (ctober 1;. Retrieved ecember !5 01!5 from %istory of Math K

%istory Modules for the Mathematics "lassroom9 http9NNhom.'i3idot.comNtheKcycloid Deidhorn5 M. :00+;. -oo3 Revie'The Person of the Millennium9 The 2ni#ue Impact of alileo on Dorld %istory. In M. A. >inocchiaro5 3he "istorian 0 6olume 79, 8ssue $  :pp. )01K )0;. Deisstein5 C. D. :n.d.;. 3autochrone .roblem . Retrieved ecember $5 01!5 from Michigan State 2niversity Bibraries9 http9NNarchive.lib.msu.eduNcrcmathNmathNmathNtNt0.htm

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