Cycloid is the curve traced by a point P on the circumference of a circle of radius a rolling along a straight line in a plane. See the figure. The parametric equation of the cycloid is

(a) Find the tangent to the cycloid at the point where \theta=\pi / 3

(b) At what points is the tangent horizontal? When is it vertical?

Solutions

Expert Solution

(a) Find the tangent to the cycloid at the point where \theta=\pi / 3

To find the tangent of the cyloid, consider equation of tangent as:

where m=dy/dx

Therefore, we can find the slop m=\frac{d y}{d x} at point of \theta=\frac{\pi}{3} as following

After that, we can find the tangent of the cyloid as following:

(b) At what points is the tangent horizontal? When is it vertical?

Therefore, the length of one arch of the cyloid is L=8a (unit of length)

Therefore, the area of one arch of the cyloid is A=3 a^{2} \pi (unit of square length)

SUN BUNRA answered 1 year ago

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