Chapter 4, Problem 22P

Arlene is to walk across a “high wire" strung horizontally between two buildings 10.0 m apart. The sag in the rope when she is at the midpoint is 10.0°, as shown in Fig. 4-47. If her mass is 50.0 kg, what is the tension in the rope at this point?

Figure 4-47
Problem 23.

To determine

Tension in the rope when the walker is midway.

Answer to Problem 22P

Tension in the rope is $1410.89\text{N}$ .

Explanation of Solution

Given:

  $d=10\text{ m}$ ......................................................................... (Distance between two buildings)

  $m=50\text{ kg}$ ........................................................................... (Mass of the high wire walker)

  $\theta =10°$ ............................................................ (Angle at the ends when walker is midway)

  $a=0{\text{m/s}}^{\text{2}}$ ............... (Acceleration in vertical direction is 0 as there is no vertical motion)

To find:

  $T=?$ ............................................................................................ (Tension along the rope)

Formula:

Newton’s second law of motion:

  ${\displaystyle \sum F=ma}$

Where, m is mass and a is acceleration

Calculation:

Free body diagram:

  

There is tension T in each rope. The component of tension in x -direction is $T\text{cos}10°$ . These cancel each other because they are equal in magnitude but opposite in direction.

The component of tension in y -direction is $T\text{sin}10°$ .

Applying Newton’s second law in the vertical direction.

  $\begin{array}{l}{\displaystyle \sum F=ma}\\ 2T\text{sin}10°–mg=ma\\ 2T\text{sin}10°–\left(50\right)\left(9.8\right)=m\left(0\right)\\ T=1410.89\text{N}\end{array}$

Conclusion: Applying Newton’s second law in vertical direction gives the tension in the rope $1410.89\text{N}$ .