Since a point in the circumference of a wheel describes, in space, a cycloid, let P, Fig. 29, be the point, referred to the axes AA' and a perpendicular at A. Let (x,y) be the coordinates of the point; then will the horizontal and vertical velocities of P be the rates of change of x and y respectively. O being the point of contact, A O=r versin. Since the cen r ter C, is vertically over O, its velocity is equal to the rate of increase of 40. In an element of time, dt, the center C will move the distance d d(r rdy √2ry—y? .. Its velocity v= the distance it moves divided by the time it moves, or v’— r versin-1y From the equation of the cycloid, x=r versin 12-√2ry—y2, r dy. Now dxdt-the velocity of the point dt curve APBA' is ds and this is the distance the point travels ds in an element of time, dt. .'. the velocity of the point, P. But ds=√ dy2+dx2= √(21 y2 2 ) o point, P (3). From (1), (2), and (3), we have, v= Hence, when a point of the circumference is in contact with the line, its veiocity is 0; when it is in the same horizontal plane as the center, its velocity horizontally and verically is the same as the velocity of the center, and when it is at the highest point, its motion is entirely horizontal, and its velocity is twice that of the center. ds dt : ds 2y2ry, we have by proportion, Since r r v'::√2ry:r. But √2ry=√(PF2+FO2)=PO. .. The velocity of P is to that of C as the chord PO is to the radius CO; that is, F and C are momentarily moving about O with equal angular velocity. (6) THE PROLATE AND CURTATE CYCLOID. 1. x=a(U—m sinf), y=a(1—-m cose) are the equations in every case. 2. The cycloid is prolate when m is >1 as AIP'B'I'A', Fig. 30, and curtate when m is <1, as PB. These equations are found thus: Let CP=ma, and [OCP=0. Then x=AN—A O— ON. But 40 arc subtended by LOCP=a0, and ON PCX sin LNPC ma sin LNPC(=LPCL=π-9)=ma sin (π—0)= ma sinė. x=a0)—ma ..x-a-ma sin0-a(0-m siné). y=PN=OC+ PC cos [NPC (=[PCL=π—() = a+ma cos (π-6)=a— ma cosa (1-m cos 9). The same reasoning applies when we assume the point to be P'. NOTE.-These curves are also called Trochoids. Prob. LX. To find the length of a Trochoid. Formula.-s=f√dx2+dy2. Since x=a(0—m sin0), dx=a(1— m cos (1)d0; and since y= π a(1-m cose), dy=am sin@do. .. s= · ƒ√dx2+dy2=aƒ * √[ (1 π m cose)2+m2sin '] do—a ƒˆ √(1+m2—2m cos 4)d0 = aƒ1* √{ (1+m)2—4m cos2 q } dq=sa(1+m) 4a I. If a fly is on the spoke of a carriage wheel 5 feet in diameter, 6 inches up from the ground, through what distance will the fly move while the wheel makes one revolution on a level plane? Let Cbe the center of the wheel, in the figure, and P the position of the fly at any time. Let OC= the radius of the carriage wheela 2 ft., PC= 2 ft., and the angle OCP =0. Let (x,y) be the coordinates of the point P. Let F, a point at the inter FIG. 80. section of the curve and AI be the position of the fly when the motion of the wheel commenced. Then since x-a(-m sine) and y-a(1-m cos 0), we have dx-a(1-m cose)de, and dy= am sine do. a2 m2 sin2 0 ... S= = ƒ√(dx2+dy2)= ƒ" √↓ {a2(1—m cost)2+ 0)d0=4af** √ √ (1+ do—as m)2—4m cos2p{dq, in which 9–10. But PC-2 ft., and π 3 5(x£!£x)' (¦ ¦ ) 3—7 ( 1£!££x) 3 (}}) 1—&c } =18.84 ft. 1.2.3. 1.3.5 3 1.2.3.4 II. .. The fly will move 18.84 ft. Prob. LXI. To find the area contained between the trochoid and its axis. π Formula.-A-fydx=2a2 )2d0-2a2 m cos 0)d0-2a2 2ƒ ̃(1—m cos 0))3¿0—2a3 (0—2m sin0+4m2 (0—sin 0 cos 0))*=2a2 (7+m2). When m=1, the curve is the cycloid and the area =37a2 as it should be. * When is replaced by (7+), this is an elliptic integral of the second kind and may be written 4a E(8,9). (c) EPITROCHOID AND HYPOTROCHOID. 1. An Epitrochoid is the roulette formed by a circle rolling upon the convex circumference of a fixed circle, and carrying a generating point either within or without the rolling circle. 2. An Hypotrochoid is the roulette formed by a circle rolling upon the concave circumference of a fixed circle, and carrying a generating point either within or without the rolling circle. 3. x=(a+b)cos 0—mbcos", y=(a+b) sin 0—mb sin b are the equations of the epitrochoids. In the figure, let C be the center of the fixed circle and O the center of the rolling circle. Let FP'Q be a portion of the curve generated by the point P' situated within the rolling circle, and let CG=x and P'G=y be the co-ordinates of the point, P'. Let A be the position of P when the rolling commences, and Draw OK perpendicular 9=LPOC through which it rolled. to CG and P'I perpendicular to OK; draw DP and DP'. Let OP'=mOP=mb and the angle ACD=0. Then x=CG=CK +KG=KC+P'I. But CK=OC cos 0=(a+b) cos and P'I= P'Ocos/OP'I=mb cos bq. Whence q q=0. .. P'I————mb cos113, and x=(a+b) cos 0a+b mbcosa‡be. y=P'G b. =IK-OK-OI. But OK OC sin / KCO= FIG. 31. (a+b)sine, and OI=OP' sin / OP'I=mb sin{ π-(p+0)}= mb sin(9+0)=mb sina‡b. b .. y=(a+b)sin 0—mb sin a+bo. If m=1, the point P' will be on the circumference of the rolling circle and will describe the curve APN which is called the Epicycloid The equations for the Epicycloid are x=(a+b)cos 9— a+b b cos- ƒ), and y=(a+b) sin 8—b sine. The equations for the b b Hypotrochoid may be obtained by changing the signs of b and mb, in the equations for the Epitrochoid... x=(a-b) cos 0+mb cos a-b a-b b b -0, and y=(a-b)sin 0-mb sin 0 are the equations for the Hypotrochoid. If m=1, the generating point is in the circumference of the rolling circle and the curve generated will be a Hypo a -b b cycloid. .. x=(a—b) cos✪ +b cos' , and y=(a-b)sin b sin b -b are the equations of the Hypocycloid . Prob. LXII. To find the length of the arc of an epitrochoid. Formula.-s= = ƒ Ndx2 + dy2 = √ √√√ { [ 2 −(a+b) sin 0+ a+b 2 m(a+b) sin- a+b1] + [(a+b) cɔs 0—m(a+b) cos2+3]* } 20 =(a+b) √ √ {1+m2—2m(sin ◊ sin a+ a cos cose 1‡bo) } do=(a+b) ƒ√(1-+m2—2m cos z 0 )d 0. This may be expressed as an elliptic integral, E(k, p), of the second kind, by substituting (7+ p) for 0, and then reducing. 26 a 2. By making m=1, we have s=(a+b)√2 S√TI— α cos) do, the length of the arc of an hypocycloid. 3. By changing sign of b, the above formula reduces to s= α (a—b) ƒ√(1+m2+2m cos)de, which is the length of the arc of an hypotrochoid. 4. By making m=1, in the last formula, we have s= a (a—b)√2 ƒ(1+cos / v) } do, which is the length of the arc of a hypocycloid. b I. A circle 2 ft. in diameter rolls upon the convex circmuference of a circle whose diameter is 6 feet. What is the length |