Question

a disc of radius r start at time t=0 moving along the positive x axis with linear speed v and angular speed omega find the x and y coordinate of the bottommost point at any time t

Answer 1

disc radius = r,   Linear velocity = v,    angular velocity = ω,   v = r ω when the disc rolls.

Let the bottommost point be P (0,0).  Let the center of circular disc be O (r,0).

Position of  'O' :   x = v t,   y = r

Point P rotates at angular velocity ω  wrt O. So its position wrt O is 
     x1 = - r Sin ωt       y1 = r - r cos ωt
                 as  ωt = angle rotated in time t.

So   absolute position of point P:  x = v t - r Sin ωt
                       y = r - r cos ωt

If we eliminate  ωt from both equations:
              r² = (x - v t)² + (y - r)²
              x = v Cos⁻¹ (1 - y/r) - r √(y(2r-y) 
             
This is called  the cycloid curve.

Answer 2

Answer:

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