Parametrization of the path described by the end of a thread

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SUMMARY

The discussion centers on the parametrization of the path traced by the end of a thread unwound from a stationary circular spool of radius R. The path is defined as r(t) = x(t)i + y(t)j, where the thread unwinds in a clockwise direction from the initial position (0, R). The participants also address the arc length s(t) of the path, emphasizing that the thread remains constrained to the xy-plane, aligning the spool's axis with the z-axis. The absence of a z(t) component is noted as a point of confusion.

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  • Understanding of parametric equations in two dimensions
  • Knowledge of circular motion and trigonometric functions
  • Familiarity with arc length calculations in calculus
  • Basic concepts of coordinate systems, particularly Cartesian coordinates
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Students and educators in mathematics, particularly those focusing on calculus and physics, as well as anyone interested in the geometric properties of circular motion and parametrization techniques.

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Homework Statement



2. Consider a stationary circular spool of thread of radius R. Assume the end of the thread is initially located at (0; R). While keeping the thread taut, the thread is unwound in a clockwise direction.

(a) Parameterize the path described by the end of the thread as r(t) = x(t)i + y(t)j. You may assume that the radius of the spool does not change as the thread is unwound.

(b) Determine the arc length, s(t), of the path traced out by the end of the thread.

Homework Equations


The Attempt at a Solution


It seems to me that it is describing the taut thread as it is more or less rotated around the spool. I am confused on why there isn't a z(t) element too.
 
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consider the axis of the spool as aligning with the z axis, and the thread constrained to the xy plane
 

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