Parametrization of the path described by the end of a thread

RIn summary, the conversation discusses a stationary circular spool of thread and the path traced by the end of the thread as it is unwound in a clockwise direction. The parameterization of the path is given as r(t) = x(t)i + y(t)j, with the assumption that the radius of the spool remains constant. The arc length, s(t), of the path is also determined. There is confusion about the lack of a z(t) element in the discussion, but it is clarified that the thread is constrained to the xy plane with z(t) = R.
  • #1
rashomon
3
0

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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  • #2
consider the axis of the spool as aligning with the z axis, and the thread constrained to the xy plane
 

Related to Parametrization of the path described by the end of a thread

1. What is parametrization?

Parametrization is the process of defining a mathematical relationship between two or more variables. In the context of a thread, parametrization involves finding an equation that describes the path of the end of the thread as it moves.

2. Why is parametrization important in studying the path of a thread?

Parametrization allows us to mathematically describe the path of the thread, which is important in understanding the behavior of the thread and predicting its movements in different situations. It also allows us to analyze and manipulate the variables that affect the thread's path.

3. How do you determine the parametrization of a thread's path?

The parametrization of a thread's path can be determined by identifying the variables that affect the thread's movement, such as tension, gravity, and friction. From there, a mathematical equation can be derived using principles of physics and calculus to describe the relationship between these variables and the position of the thread's end.

4. Can parametrization be used to predict the path of a thread in different scenarios?

Yes, parametrization allows us to manipulate the variables in the equation and make predictions about how the thread will behave in different situations. This can be useful in applications such as robotics or sewing, where understanding the path of a thread is important.

5. Are there any limitations to parametrization in describing the path of a thread?

While parametrization can accurately describe the path of a thread in many scenarios, there are limitations. Factors such as the thread's thickness, stiffness, and elasticity may not be fully accounted for in the parametrization equation. Additionally, external factors such as air resistance may also affect the thread's path and may be difficult to incorporate into the equation.

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