Chapter 2, Problem 37A

Sneezlee is supported by two thin wires. Is the tension in each wire less than, equal to, or more than half his weight? Use the parallelogram rule to defend your answer.

To determine

To identify: Whether the tension in each wire is less than, equal to or more than half the weight of bird.

Answer to Problem 37A

Tension in each wire is greater than half the weight of bird.

Explanation of Solution

Given:

A bird sitting on a rod supported by two ropes as shown below.

Formula used:

Total tension upwards equals the total weight downwards.

Calculation:

Let $W$ be the weight of bird and $T_1$ and $T_2$ are tensionsof the ropes. ?is the angle that each part of the wire makes with the horizontal. In equilibrium,

  $T_{1}sin\theta +T_{2}sin\theta =W$

As bird is at equilibrium position on the bench, so tension in both ropes will be equal.

Thus,

  $T_1=T_2=T$

So,

  $\begin{array}{l}Tsin\theta +Tsin\theta =W\\ 2T\sin \theta =W\\ T=\frac{1}{2\sin \theta }W\end{array}$

If ropes are at $90°$ , the tension in each rope will be equal to half the weight of bird. Since the rope is slightly inclined, so angle will be less than $90°$ and value of $\sin \theta$ will be smaller than 1. So, tension in this case will greater than half the weight of bird.