Q.64 - Calculus

Question

The involute of a circle is the curve described by the endpoint P of a thread as it unwinds from a fixed circular spool. For simplicity suppose that the radius of the circular spool is r and that when the spool is placed with its center at the origin, the point P starts at (r, 0). Assume that the thread is unwinding counterclockwise. Find parametric equations for the point P. (Hint: If the string is taut at all times as it unwinds, the length of segment PT is rθ.) 

Step-by-Step Solution

Answer

As a conclusion, the solution is  $P(x,y)=(\cos \theta +r\theta \sin \theta ,\sin \theta -r\theta \cos \theta )$

Step: 1 Given information

Consider the end point of a circular pool to be an involute of a circle.

Step 2: Calculation

Assume that the diameter of the spool is $r$and when the spool is placed with its center at the origin, the point $P$starts at$(r, 0)$.

Assume the thread is unwinding in the opposite direction.

The goal is to discover the point's parametric equations. $P(x, y)$

The length of the line segment is now determined.$T P$ is just the amount of string unwound. which is equal to .

We can find the coordinates of  $P$by starting with the coordinates of$T$ and subtracting the vertical and horizontal components of the line segment$T P$.

Its coordinates of $T$are $(r\cos \theta ,r\sin \theta )$it lies on the circle.

Draw a horizontal line through it.$T$ which is on the circle.

Step:3 Further calculation

This horizontal line makes an angle with TP. is $\frac{\pi }{2}-\theta .$

The horizontal elements are $r\theta \cos \left(\frac{\pi }{2}-\theta \right).$

Then the horizontal component is $r\theta \sin \theta .$

The vertical component of $T P$is $-r\theta \sin \left(\frac{\pi }{2}-\theta \right)$

Then the vertical component is $-r\theta \cos \theta .$

The coordinates of $P(x, y)$

$x=\cos \theta +$Horizontal component

$x=\cos \theta +r\theta \sin \theta$

$y=\sin \theta +$vertical component

$y=\sin \theta -r\theta \cos \theta$[since the horizontal component~ s-$r\theta \cos \theta$]

Thus, the required parametric equations are $P(x,y)=(\cos \theta +r\theta \sin \theta ,\sin \theta -r\theta \cos \theta )$