An element ds of arc length, in terms of dx and dy, is given by the theorem of Pythagoras: ds=((dx)2+(dy)2))1/2 or, since x and y are given by the parametric Equatio...An element ds of arc length, in terms of dx and dy, is given by the theorem of Pythagoras: ds=((dx)2+(dy)2))1/2 or, since x and y are given by the parametric Equations 19.1.1 and 19.1.2, by 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.
When the circle has rolled through an angle 2θ, the centre of the circle has moved to the right by a horizontal distance 2aθ, while the horizontal distance of the point P from the...When the circle has rolled through an angle 2θ, the centre of the circle has moved to the right by a horizontal distance 2aθ, while the horizontal distance of the point P from the centre of the circle is asin2θ and the vertical distance of the point P below the centre of the circle is acos2θ.
These equations are the parametric equations of a cycloid. (For more on the cycloid, see Chapter 19 of the Classical Mechanics notes in this series.) The motion is a circular motion in which the centr...These equations are the parametric equations of a cycloid. (For more on the cycloid, see Chapter 19 of the Classical Mechanics notes in this series.) The motion is a circular motion in which the centre of the circle drifts (hence the subscript D) in the x-direction at speed VD.
