Calc1231W13T2 - Arc length

Sunday, 21 October 2018

5:44 PM

Untitled picture.png Machine generated alternative text: then the arc length ofC is defined to be: As a first example, let's find the length of the curve y Recall that if C is a curve in the plane expressed in parametric form as sin(3t), y(t) — sin(3t)_ Then x from x • if x = 0 then t • if x = 1 then t Hence the arc length is Pi,'6 b = J (x' + (y'(t)) dt. 1 0 to 1. To make things interesting we'll use the parameterisation x(t) sqä(2) Untitled picture.png The formula for the arc length of a curve is slightly different for curves specified in parametric polar form: C : r = f(9) where a 9 < b Then the arc length of C is This is also often written as b b 2 dr Consider the curve C : r = 9 sin(9) where 0 < 9 < Then the arc length of this curve is I circle of radius Note: Maple syntax for is pi _ (19. O If we graph this function; then our answer is unsurprising since it's a Untitled picture.png Consider the curve which looks like -20 The arc length of C is 27T Note: Maple syntax for is pi _ 392 where 0 < 9 < 2K 20 40 2 60 80 100 120 Untitled picture.png Machine generated alternative text: XP alf(x, t) YP diff(y, t) + yp2), t = 0 x sin(3 t) y sin(3 t) Y --3 cos(3t)
Untitled picture.png Consider the curve which looks like -20 The arc length of C is 27T Note: Maple syntax for is pi _ 392 where 0 < 9 < 2K 20 40 2 60 80 100 120 Untitled picture.png The cardioid is the curve The graph of the cardioid is: The length of C is given by the formula 27T where 1 + sin(9) where 0 9 < 2K 2K 2 dr To evaluate this integral, we could use a trick: multiply top and bottom by 1 — sin(9) Evaluating we get 8 Note: because most keyboards lack a 9 button, you are asked to specify p as a function of t. Untitled picture.png r 382 r = 382 + theta)2), theta—0 2 pi)
Untitled picture.png The cardioid is the curve The graph of the cardioid is: The length of C is given by the formula 27T where 1 + sin(9) where 0 9 < 2K 2K 2 dr To evaluate this integral, we could use a trick: multiply top and bottom by 1 — sin(9) Evaluating we get 8 Note: because most keyboards lack a 9 button, you are asked to specify p as a function of t. Untitled picture.png A train wheel of radius 2 feet moves one complete revolution along a straight track without slipping Initially the point A on the wheel is touching the track. 4 2 3TT x 4TT The point A touches the track again after the train has moved 4*Pi The locus of the point A is called the cycloid; it can be described parametrically as the curve O ofthe train wheel A: [2(t — sin(t)), — cos(t))] where 0 t 2K Hence the length of the curve A is 16 O feet. The cycloid is very famous historically; it has a number of remarkable properties. Among them is the tautochrone property that the time taken for an object to roll down the side of an inverted cycloid is independent of the starting point Note: for those who are not familiar with the aæh-aÉ imperial system, 1 foot is approximately 0.3048 metres_
Untitled picture.png A train wheel of radius 2 feet moves one complete revolution along a straight track without slipping Initially the point A on the wheel is touching the track. 4 2 3TT x 4TT The point A touches the track again after the train has moved 4*Pi The locus of the point A is called the cycloid; it can be described parametrically as the curve O ofthe train wheel A: [2(t — sin(t)), — cos(t))] where 0 t 2K Hence the length of the curve A is 16 O feet. The cycloid is very famous historically; it has a number of remarkable properties. Among them is the tautochrone property that the time taken for an object to roll down the side of an inverted cycloid is independent of the starting point Note: for those who are not familiar with the aæh-aÉ imperial system, 1 foot is approximately 0.3048 metres_ Untitled picture.png int (sqrt( diff (2* (t—sin (t)) , t) + diff (2* (I—cos (t)) , 16 t) .2*Pi) ;

 

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