Q23. Arlene is to walk across a 鈥渉i... [FREE SOLUTION]

Chapter 4: Q23. (page 102)

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

Short Answer

Expert verified

The obtained value of tension is $T = 1412.33 N$ .

Step by step solution

Step 1. Calculate the sum of the vertical forces

For this condition, assume the point in the rope directly below Arlene. Also, the rope is massless. Three forces are acting on her, the weight of the body, tension in the rope toward the right and the left point of joining.

Given data:

The distance between two buildings is $d = 10 m$ .

The sag in the rope is of angle $胃 = 10 掳$ .

The mass of Arlene is $m = 50 kg$ .

The free body diagram of the girl is as follows:

The relation to calculate the vertical forces is given by:

$\begin{matrix} 危 F_{v} = 0 \\ T \sin 胃 + T \sin 胃 - W = 0 \end{matrix}$

Here, T is the tension force, and W is the weight.

Step 2. Determine the tension in the rope

On plugging the values in the above relation, you get:

$\begin{matrix} T \sin 10 掳 + T \sin 10 掳 - m g = 0 \\ T \sin 10 掳 + T \sin 10 掳 - (50 kg) (9.81 m / s^{2}) = 0 \\ 2 T \sin 10 掳 = (490.5 kg 路 m / s^{2}) \\ T = 1412.33 N \end{matrix}$