How Does the Point of Tangency Move in Circular Motion?

In summary, the conversation explains how to calculate the position and direction of a string that is unwound from a bobbin moving around a circle of radius 10. The point of tangency of the string moves at 2π radians per second and is parameterized as (10 cos 2πt, 10 sin 2πt). The end of the string relative to the point of tangency moves a length of 20πt and points in the direction of (sin 2πt, -cos 2πt). This is determined by computing the derivatives of the components of the parameterization, which represent the velocity vector.
  • #1
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Homework Statement
Imagine a string (of negligible thickness) unwinding from a fixed circular bobbin of radius 10, so that the string unwound is always tangent to the bobbin, as in the picture attached. Assume that the string unwinds at the constant rate of one full loop of string per second. Determine parametric equations for the curve traced out by the end of the string, as shown by the dark curve in the picture.
Relevant Equations
parametric equations
picture.jpg

Solution:
The point of tangency of the string moves around the circle at ##2\pi## radians per second. First, we compute the position of the point of tangency of the string with the bobbin. Because this is simply a revolution around a circle of radius 10, the parameterization of the point of tangency is ##(10 \cos 2\pi t, 10 \sin 2\pi t)##. Now we compute the position of the end of the string relative to the point of tangency with the bobbin. After t seconds have passed, the point of tangency has moved a length of ##20\pi t## around the bobbin. Thus, a length of ##20\pi t## of string has been unwound. The direction that the string points is tangent to the bobbin. By computing the derivatives of the components in ##(10 \cos 2\pi t, 10 \sin 2\pi t)##, we realize that the direction the string points in is the direction of ##(\sin 2\pi t, – \cos 2\pi t)##.

I don't understand the bolded part above. Could anyone explain it? Thanks.
 
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  • #2
Are you confused about the claim that if the coordinates are (x(t),y(t)) then (x'(t),y'(t)) is the velocity vector?
 

What are parametric equations?

Parametric equations are a set of equations that express the coordinates of a point in terms of one or more parameters. These equations are commonly used to describe the motion of a point in a plane or in space.

Why are parametric equations useful?

Parametric equations allow us to easily describe complex curves and shapes by breaking them down into simpler equations. They also allow us to model and analyze motion and change in a more intuitive way.

How do you find parametric equations?

The process of finding parametric equations involves identifying the parameters, determining the coordinates of the point in terms of those parameters, and then eliminating the parameters to obtain the final equations. This can be done algebraically or graphically.

What are some real-life applications of parametric equations?

Parametric equations are used in various fields such as physics, engineering, and computer graphics. They can be used to model the motion of objects, design complex shapes and curves, and create animations and simulations.

What are the differences between parametric equations and Cartesian equations?

The main difference is that parametric equations use parameters to describe a point's coordinates, whereas Cartesian equations use variables. Parametric equations also allow for more flexibility in describing curves and shapes, while Cartesian equations are limited to straight lines and conic sections.

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