Q.62 - Calculus

Question

A trochoid is a generalization of a cyclod in which the point tracing the path is on a spoke of the wheel, instead of on the circumference. Thus, if the radius of the wheel is constant is, the point is $k$units from the center of the wheel, such that either $k<r$or (when you can think of the point being on a flange extending the radius of the wheel. This case occurs as train wheels roll, since there is an extension of each wheel beyond the portion of the wheel rolling on the track. Find parametric equations for the trochoid.

Step-by-Step Solution

Answer

As a conclusion, the curve's length is equal to  $2{\pi }^2${ or } $19.76$

Step: 1 Given information

A point of a trochoid in which the radius of wheel is $r$and the point is $k$units from the center of the wheel.

Step 2: Calculation

The goal is to find the trochoid's parametric equations.

A trochoid is a generalization of a cycloid in which the path is traced on the spokes of the wheel rather than the circumference.

From the diagram $D$is a point on the circle and $\theta$is the angle.

The point $p$started initially from the origin. And it has travelled by an angle of$\theta$.

So $PR=OR=r\theta$

To find $x, y$ the coordinates of $D$

From the triangle $C D Q$

$$\begin{gathered} CD=d \\ CQ=d\cos \theta ,DQ=d\sin \theta \end{gathered}$$

Step:3: Further Calculation

The $x$coordinate is given by $D=O R-D Q$

$x=r\times \theta -d\sin \theta$

The $x$coordinate is given by $D=C R-Q R$

$y=r-d\cos \theta$

Thus, the parametric equations are$x=r\theta -d\sin \theta ,y=r-d\cos \theta$.

Therefore, the required parametric equations are$x=r\theta -d\sin \theta ,y=r-d\cos \theta$.

When $\theta =2\pi ,$

$(x,y)=(\cos \theta +\theta \sin \theta ,\sin \theta -\theta \cos \theta )$

Then,

$$\begin{gathered} (x,y)=(\cos 2\pi +2\pi \sin 2\pi ,\sin 2\pi -2\pi \cos 2\pi ) \\ (x,y)=(1+0,0-2\pi ·1) \\ (x,y)=(1,-2\pi ) \\ (x,y)=(1,-6.2) \end{gathered}$$