Chapter 8.2, Problem 74E

Solution

(a.)

To Prove: The plot of the velocity versus position (distance) is an ellipse.

It has been shown that the plot of the velocity versus position (distance) is an ellipse.

Given:

A pendulum that swings toward and away from a motion detector has a distance (in feet) from the detector of $x\left(t\right)=5+3\sin \left(\pi t+\frac{\pi }{2}\right)$ and a velocity (in ft/sec) of $y\left(t\right)=3\pi \cos \left(\pi t+\frac{\pi }{2}\right)$ , where $t$ represents time (in seconds).

Concept used:

Parametric equations $x\left(t\right)$ and $y\left(t\right)$ can be simplified to obtain an equation in $x$ and $y$ by eliminating the parameter $t$ .

Calculation:

The given parametric equations are $x\left(t\right)=5+3\sin \left(\pi t+\frac{\pi }{2}\right)$ and $y\left(t\right)=3\pi \cos \left(\pi t+\frac{\pi }{2}\right)$ .

Simplifying,

  $\frac{x-5}{3}=\sin \left(\pi t+\frac{\pi }{2}\right)$ ,

  $\frac{y}{3\pi }=\cos \left(\pi t+\frac{\pi }{2}\right)$

Squaring both sides of each equation,

  $\frac{{\left(x-5\right)}^2}{3^2}={\sin }^2\left(\pi t+\frac{\pi }{2}\right)$ ,

  $\frac{y^2}{{\left(3\pi \right)}^2}={\cos }^2\left(\pi t+\frac{\pi }{2}\right)$

Adding both equations,

  $\frac{{\left(x-5\right)}^2}{3^2}+\frac{y^2}{{\left(3\pi \right)}^2}={\sin }^2\left(\pi t+\frac{\pi }{2}\right)+{\cos }^2\left(\pi t+\frac{\pi }{2}\right)$

Using trigonometric identity; ${\sin }^2\theta +{\cos }^2\theta =1$ ,

  $\frac{{\left(x-5\right)}^2}{3^2}+\frac{y^2}{{\left(3\pi \right)}^2}=1$

This is the equation of an ellipse in standard form.

This shows that the plot of the velocity versus position (distance) is an ellipse.

Conclusion:

It has been shown that the plot of the velocity versus position (distance) is an ellipse.

(b.)

To Describe: The motion of the pendulum.

The motion of the given pendulum has been described.

Given:

A pendulum that swings toward and away from a motion detector has a distance (in feet) from the detector of $x\left(t\right)=5+3\sin \left(\pi t+\frac{\pi }{2}\right)$ and a velocity (in ft/sec) of $y\left(t\right)=3\pi \cos \left(\pi t+\frac{\pi }{2}\right)$ , where $t$ represents time (in seconds).

Concept used:

Parametric equations $x\left(t\right)$ and $y\left(t\right)$ can be simplified to obtain an equation in $x$ and $y$ by eliminating the parameter $t$ .

Calculation:

The given parametric equations are $x\left(t\right)=5+3\sin \left(\pi t+\frac{\pi }{2}\right)$ and $y\left(t\right)=3\pi \cos \left(\pi t+\frac{\pi }{2}\right)$ .

Put $t=0$ in $x\left(t\right)=5+3\sin \left(\pi t+\frac{\pi }{2}\right)$ and $y\left(t\right)=3\pi \cos \left(\pi t+\frac{\pi }{2}\right)$ to get,

  $x\left(0\right)=8$ ,

  $y\left(0\right)=0$

Put $t=\frac{1}{2}$ in $x\left(t\right)=5+3\sin \left(\pi t+\frac{\pi }{2}\right)$ and $y\left(t\right)=3\pi \cos \left(\pi t+\frac{\pi }{2}\right)$ to get,

  $x\left(\frac{1}{2}\right)=5$ ,

  $y\left(\frac{1}{2}\right)=-3\pi$

Put $t=1$ in $x\left(t\right)=5+3\sin \left(\pi t+\frac{\pi }{2}\right)$ and $y\left(t\right)=3\pi \cos \left(\pi t+\frac{\pi }{2}\right)$ to get,

  $x\left(1\right)=2$ ,

  $y\left(1\right)=0$

Put $t=\frac{3}{2}$ in $x\left(t\right)=5+3\sin \left(\pi t+\frac{\pi }{2}\right)$ and $y\left(t\right)=3\pi \cos \left(\pi t+\frac{\pi }{2}\right)$ to get,

  $x\left(\frac{3}{2}\right)=5$ ,

  $y\left(\frac{3}{2}\right)=3\pi$

So, the pendulum starts from a position $8$ feet away from the motion detector with zero initial velocity and achieves the maximum velocity of $3\pi$ ft/sec when it is $5$ feet away from the motion detector. Then, its velocity decreases until it reaches $2$ feet away from the motion detector, when its velocity becomes zero. The pendulum then reverses direction and swings away from the motion detector and again achieves a maximum velocity of $3\pi$ ft/sec when it is $5$ feet away from the motion detector, until it reaches its starting point where its velocity again becomes zero.

Conclusion:

The motion of the given pendulum has been described.