Chapter 6.3, Problem 108E

(a)

To determine

To find: The tension T in the rope and the magnitude of u as function of $\theta$ .

Answer to Problem 108E

  $T=\sec \theta$

  $\left|u\right|=\tan \theta$

Explanation of Solution

Given: A tetherball weighing 1 pounds is called outwards from the pole by a horizontal force u until the rope makes an angle of $\theta$ degrees with the pole as shown in figure.

  

First draw FBD of the given diagram.

  

The horizontal and vertical component of FBD is in equilibrium.

  $\begin{array}{l}\therefore T\sin \theta =u\\ T\cos \theta =1\end{array}$

The function of tension in the form of $\theta$

  $T=\sec \theta$

Domain, $D\in \left(0,{90}^{\circ }\right)$

The magnitude of u, $u=T\sin \theta$

Where, $T=\sec \theta$

  $\left|u\right|=\tan \theta$

  $D\in \left(0,{90}^{\circ }\right)$

(b)

To determine

To graph: The two functions form $0^{\circ }\le \theta \le {60}^{\circ }$

Answer to Problem 108E

  

Explanation of Solution

Given: A tetherball weighing 1 pounds is called outwards from the pole by a horizontal force u until the rope makes an angle of $\theta$ degrees with the pole as shown in figure.

  

The function of tension in the form of $\theta$

  $T=\sec \theta$

The function of magnitude of u, $\left|u\right|=\tan \theta$

  

(c)

To determine

To compare: The both functions as $\theta$ increases.

Answer to Problem 108E

Both are increasing functions.

Explanation of Solution

Given: A tetherball weighing 1 pounds is called outwards from the pole by a horizontal force u until the rope makes an angle of $\theta$ degrees with the pole as shown in figure.

  

The function of tension in the form of $\theta$ , $T=\sec \theta$

The function of magnitude of u, $\left|u\right|=\tan \theta$

As shown on graph both are increasing as $\theta$ increases.

But the increasing rate of magnitude of u is larger than increasing rate of tension T .