Cycloid: Derivation

A: The Original Circle

Building from the homework we can get a circle starting from the bottom and moving clockwise by using:[table][tr][td][math]t[/math][/td][td][math]0[/math][/td][td][math]\frac{\pi}{2}[/math][/td][td][math]\pi[/math][/td][td][math]\frac{3\pi}{2}[/math][/td][td][math]2\pi[/math][/td][/tr][tr][td][math]x[/math][/td][td][math]0[/math][/td][td][math]-1[/math][/td][td][math]0[/math][/td][td][math]1[/math][/td][td][math]0[/math][/td][/tr][tr][td][math]y[/math][/td][td][math]-1[/math][/td][td][math]0[/math][/td][td][math]1[/math][/td][td][math]0[/math][/td][td][math]-1[/math][/td][/tr][/table][br]So we can use the equations:[br][math]x_t=-\sin t[/math][br][math]y_t=-\cos t[/math][br][br]Click "Animate" to see the point [math]P[/math] trace out the path described around the circle.

B. Vertical Translation

If we want to translate the whole circle up [math]1[/math] unit, we can add 1 to [math]y_t[/math]:[br][math]x_t=-\sin t[/math][br][math]y_t=-\cos t+1[/math][br]Check box B, then Animate.[br]

C. Horizontal Translation by a Constant

If we want to translate the whole circle right [math]3[/math] units, for example, we can add [math]3[/math] to [math]x_t[/math]:[br][math]x_t=-\sin t+3[/math][br][math]y_t=-\cos t+1[/math][br]Check box C, then Animate.[br]

D. Horizontal Translation by a Variable

If we want to translate the whole circle right [math]t[/math] units, for example, we can add [math]t[/math] to [math]x_t[/math]:[br][math]x_t=-\sin t+t[/math][br][math]y_t=-\cos t+1[/math][br]Check box D, then Animate.

Observations

[list][*]Now the graph is being moved by a variable amount. Our original circle is being translated horizontally, but at the same time, the point [math]P[/math] is moving around the circle. When [math]t=2\pi[/math], the point [math]P[/math] will have gone all the way around the circle, which will now be [math]2\pi[/math] units to the right of where it started.[/*][*]Notice that [math]2\pi[/math] is exactly the circumference of the circle.[/*][*]Click “Show Circumference” and reset the animation to see how the circumference ‘unrolls’ along the x-axis.[/*][*]Every time the wheel makes one full rotation, the distance it moves equals the circumference.[/*][/list]