CLASS19 - 1 CHAPTER IXX THE CYCLOID 19.1 Introduction...

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1 CHAPTER IXX THE CYCLOID 19.1 Introduction 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 x y FIGURE IXX.1 Let us set up a coordinate system O xy , and a horizontal straight line y = 2 a . We imagine a circle of diameter 2 a between the x -axis and the line y = 2 a , and initially the lowest point on the circle, P, coincides with the origin of coordinates O. We now allow the circle to roll counterclockwise without slipping on the line y = 2 a , so that the centre of the circle moves to the right. As the circle rolls on the line, the point P describes a curve, which is known as a cycloid. When the circle has rolled through an angle 2 θ , the centre of the circle has moved to the right by a horizontal distance 2 a θ , while the horizontal distance of the point P from the centre of the circle is a sin , 2 θ and the vertical distance of the point P below the centre of the circle is a cos . 2 θ Thus the coordinates of the point P are x a = + ( sin ), 22 θ θ 19.1.1 and ya = ( cos ). 12 θ 19.1.2 These are the parametric equations of the cycloid. Equation 19.1.2 can also be written = 2 2 sin . θ 19.1.3 P P 2 θ ψ A
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2 Exercise : When the x -coordinate of P is 2.500 a , what (to four significant figures) is its y - coordinate? Solution : We have to find 2 θ by solution of 22 2 5 θ θ + = sin . . By Newton-Raphson iteration (see Chapter 1 of the Stellar Atmospheres notes in this series) or otherwise, we find 2 θ = 0.931 599 201 radians, and hence y = 0.9316 a . 19.2 Tangent to the Cycloid The slope of the tangent to the cycloid at P is dy / dx , which is equal to (/) / , dy d dx d θθ and these can be obtained from equations 19.1.1 and 19.1.2. Exercise : Show that the slope of the tangent at P is tan θ . That is to say, the tangent at P makes an angle θ with the horizontal. Having done that, now consider the following: Let A be the lowest point of the circle. The angle ψ that AP makes with the horizontal is given by tan . ψ θ = y xa 2 Exercise : Show that ψ = θ . Therefore the line AP is the tangent to the cycloid at P; or the tangent at P is the line AP. 19.3 The Intrinsic Equation to the Cycloid An element ds of arc length, in terms of dx and dy , is given by the theorem of Pythagoras: () () () , 2 / 1 2 2 dy dx ds + = or, since x and y are given by the parametric equations 19.1.1 and 19.1.2, by . 2 / 1 2 2 θ θ + θ = d d dy d dx ds And of course we have just shown that the intrinsic coordinate ψ (i.e. the angle that the tangent to the cycloid makes with the horizontal) is equal to θ . Exercise : Integrate ds (with initial condition s = 0, θ = 0) to show that the intrinsic equation to the cycloid is s a = 4 sin . ψ 19.3.1 Also, eliminate ψ (or θ ) from equations 19.3.1 and 19.1.2 to show that the following relation holds between arc length and height on the cycloid: sa y 2 4 = . 19.3.2
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3 19.4 Variations In sections 19.1,2,3, we imagined that the cycloid was generated by a circle that was rolling counterclockwise along the line y = 2 a . We can also imagine variations such as the circle rolling clockwise along y = 0, or we can start with P at the top of the circle rather than at the bottom. I summarise in this section four variations.
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CLASS19 - 1 CHAPTER IXX THE CYCLOID 19.1 Introduction...

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