String Wound Around a Circle

Question

A string is wound clockwise around the circle of radius centered at the origin O; the initial position of the end P of the string is ($a$, 0). Unwind the string, always pulling it taut (so it stays tangent to the circle). Write parametric equations for the motion of P.

When I solved this I considered a point Q on the circumference of the circle to be in the fourth quadrant which gave me $ \vec {OQ}= ( a~ cos \theta, -a ~ sin \theta)$ considering $\theta$ is the angle swept out

and since $ \vec {QP}$ is perpendicular to $ \vec{OQ}$ I got $ \vec {QP} = (-a ~\theta sin\theta,-a ~\theta cos\theta) $ which gives

$$\vec {OP} = \vec {OQ} + \vec {QP}= a ~(cos \theta- \theta sin \theta,-sin \theta- \theta cos \theta ) $$

But in the solution point Q is considered in the first quadrant and hence the answer is

$$ \vec {OP} = a~(cos \theta + \theta sin \theta,sin \theta- \theta cos \theta ) $$

where is my understanding of the problem going wrong ?

Answer 1

Initially a long string is wound on a cylinder/circle in clockwise rotation. After that unwinding begins from $(a,0)$ at P in the first quadrant.

The problem given is for small angle $\theta $ increments in the first quadrant for anticlockwise rotation of the angle subtended at origin as well as the taut string segment.

The parametrization should be

$$ \vec {OQ} = a~(cos \theta + \theta sin \theta,~sin \theta - \theta cos \theta ) $$

You considered instead clockwise rotation.

So substituting $-\theta $ in place of $+\theta $ you should get for unwinding in second quadrant:

$$ \vec {OQ} = a~(cos \theta + \theta sin \theta, ~ -sin \theta + \theta cos \theta ) $$

circ = ParametricPlot[{Cos[t], Sin[t]}, {t, 0, 2 Pi}]
involuCCW = 
  ParametricPlot[{Cos[t] + t Sin[t], Sin[t] - t Cos[t]}, {t, 0, 
    Pi /2}];
involuCW = 
  ParametricPlot[{Cos[t] + t Sin[t], -Sin[t] + t Cos[t]}, {t, 0, 
    Pi /2}];
Show[{circ, involuCCW, involuCW}, PlotRange -> All, 
 GridLines -> Automatic]