An object, P, moves along a horizontal rail as seen in the figure below. The path of the rail is given by the function y= f(x). As the object, P, moves along the rail it rotates about its centroid, so that its front always faces a point Q. When P is aligned with the y-axis the rotation angle is zero. If the coordinate of Q is (100, 200), show that the equation for the rotation matrix, R, of P is given by:  

θ= arccos⎛⎝⎜200−f(x)(200 −f(x))2 + (100−x)2−−−−−−−−−−−−−−−−−−−−−−√⎞⎠⎟
𝜃= arccos⁡(200−f(x)(200 −f(x))2 + (100−x)2)
 
   

R=(cosθsinθsin−θcosθ)
R=cos𝜃sin−𝜃sin𝜃cos𝜃
 
 

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