Chapter 1.4, Problem 35E

(a)

To determine

To find: The parameterization to model the motion of a particle with initial point $\left(a,0\right)$ and traces the circle $x^2+y^2=a^2$ , $a>0$ once clockwise.

Answer to Problem 35E

The possible parameterization is $x=a\cos t$ and $y=-a\sin t$ in the interval $0\le t\le 2\pi$.

Explanation of Solution

Given information: The particle starts motion from $\left(a,0\right)$ and traces the circle $x^2+y^2=a^2$ , $a>0$ once clockwise.

Calculation:

The standard parameterization for a circle with radius $a$ is given by $x=a\cos t$ and $y=a\sin t$.

Substitute $-t$ for $t$ in the equations $x=a\cos t$ and $y=a\sin t$.

  $\begin{array}{c}x=a\cos \left(-t\right)\\ =a\cos t\\ y=a\sin \left(-t\right)\\ =-a\sin t\end{array}$

As it is given that the particle start at $\left(a,0\right)$ . So, substitute $t=0$.

  $\begin{array}{c}x=a\cos \left(0\right)\\ =a\\ y=-a\sin \left(0\right)\\ =0\end{array}$

The particle will complete one cycle in an interval of $2\pi$ . So, substitute an end point at $0+2\pi =2\pi$.

  $\begin{array}{c}x=a\cos \left(2\pi \right)\\ =a\\ y=-a\sin \left(2\pi \right)\\ =0\end{array}$

Therefore, the possible parameterization is $x=a\cos t$ and $y=-a\sin t$ in the interval $0\le t\le 2\pi$.

(b)

To determine

To find: The parameterization to model the motion of a particle that starts at $\left(a,0\right)$ and traces the circle $x^2+y^2=a^2$ , $a>0$ once counterclockwise.

Answer to Problem 35E

The possible parameterization is $x=a\cos t$ and $y=a\sin t$ in the interval $0\le t\le 2\pi$.

Explanation of Solution

Given information: The particle starts at $\left(a,0\right)$ and traces the circle $x^2+y^2=a^2$ , $a>0$ once counterclockwise.

Calculation:

The standard parameterization for a circle with radius $a$ is given by $x=a\cos t$ and $y=a\sin t$ by which the particle move counterclockwise.

As it is given that the particle start at $\left(a,0\right)$ . So, substitute $t=0$.

  $\begin{array}{c}x=a\cos \left(0\right)\\ =a\\ y=a\sin \left(0\right)\\ =0\end{array}$

The particle will complete one cycle in an interval of $2\pi$ . So, substitute an end point at $0+2\pi =2\pi$.

  $\begin{array}{c}x=a\cos \left(2\pi \right)\\ =a\\ y=a\sin \left(2\pi \right)\\ =0\end{array}$

Therefore, the possible parameterization is $x=a\cos t$ and $y=a\sin t$ in the interval $0\le t\le 2\pi$.

(c)

To determine

To find: The parameterization to model the motion of a particle that starts at $\left(a,0\right)$ and traces the circle $x^2+y^2=a^2$ , $a>0$ twice clockwise.

Answer to Problem 35E

The possible parameterization is $x=a\cos t$ and $y=-a\sin t$ in the interval $0\le t\le 4\pi$.

Explanation of Solution

Given information: The particle starts at $\left(a,0\right)$ and traces the circle $x^2+y^2=a^2$ , $a>0$ twice clockwise.

Calculation:

The standard parameterization for a circle with radius $a$ is given by $x=a\cos t$ and $y=a\sin t$ by which the particle move twice clockwise.

Substitute $-t$ for $t$ in the equations $x=a\cos t$ and $y=a\sin t$.

  $\begin{array}{c}x=a\cos \left(-t\right)\\ =a\cos t\\ y=a\sin \left(-t\right)\\ =-a\sin t\end{array}$

As it is given that the particle start at $\left(a,0\right)$ . So, substitute $t=0$.

  $\begin{array}{c}x=a\cos \left(0\right)\\ =a\\ y=-a\sin \left(0\right)\\ =0\end{array}$

The particle will complete two cycle in an interval of $2\cdot 2\pi =4\pi$ . So, substitute an end point at $0+4\pi =4\pi$.

  $\begin{array}{c}x=a\cos \left(4\pi \right)\\ =a\\ y=-a\sin \left(4\pi \right)\\ =0\end{array}$

Therefore, the possible parameterization is $x=a\cos t$ and $y=-a\sin t$ in the interval $0\le t\le 4\pi$.

(d)

To determine

To find: The parameterization to model the motion of a particle that starts at $\left(a,0\right)$ and traces the circle $x^2+y^2=a^2$ , $a>0$ twice counterclockwise.

Answer to Problem 35E

The possible parameterization is $x=a\cos t$ and $y=a\sin t$ in the interval $0\le t\le 4\pi$.

Explanation of Solution

Given information: The particle starts at $\left(a,0\right)$ and traces the circle $x^2+y^2=a^2$ , $a>0$ twice counterclockwise.

Calculation:

The standard parameterization for a circle with radius $a$ is given by $x=a\cos t$ and $y=a\sin t$ by which the particle move twice clockwise.

As it is given that the particle start at $\left(a,0\right)$ . So, substitute $t=0$.

  $\begin{array}{c}x=a\cos \left(0\right)\\ =a\\ y=a\sin \left(0\right)\\ =0\end{array}$

The particle will complete two cycle in an interval of $2\cdot 2\pi =4\pi$ . So, substitute an end point at $0+4\pi =4\pi$.

  $\begin{array}{c}x=a\cos \left(4\pi \right)\\ =a\\ y=a\sin \left(4\pi \right)\\ =0\end{array}$

Therefore, the possible parameterization is $x=a\cos t$ and $y=a\sin t$ in the interval $0\le t\le 4\pi$.