# The distance between the two telephone poles is 50.0 m. When a 1.00 kg bird lands on the...

## Question:

The distance between the two telephone poles is 50.0 m. When a 1.00 kg bird lands on the telephone wire midway between the poles, the wire sags 0.200 m. Draw a free-body diagram of the bird. How much tension does the bird produce in the wire? Ignore the weight of the wire.

## Force:

When three different forces act on a single point where one of the force and angles between these forces is given, then the other two forces can be determined using Lami's theorem or analytical method.

## Answer and Explanation:

**Given Data**

- The distance between the two poles is; {eq}d = 50.0\;m {/eq}

- The mass of bird is; {eq}m = 1\;{\rm{kg}} {/eq}

- The sagging of wire is; {eq}y = 0.2\;{\rm{m}} {/eq}

Draw a free body diagram of the bird.

Find the angle {eq}\theta {/eq} from the figure.

{eq}\begin{align*} \tan \theta &= \dfrac{{CD}}{{DB}}\\ \tan \theta & = \dfrac{{{\rm{0}}{\rm{.2}}\;{\rm{m}}}}{{{\rm{25}}\;{\rm{m}}}}\\ \theta & = {\tan ^{ - 1}}\left( {\dfrac{{{\rm{0}}{\rm{.2}}\;{\rm{m}}}}{{{\rm{25}}\;{\rm{m}}}}} \right)\\ \theta & \approx 0.458^\circ \end{align*} {/eq}

Find the angle {eq}\phi {/eq} from the figure.

{eq}\phi = 90^\circ - \theta \\ \phi = 90^\circ - 0.458^\circ = 89.542^\circ {/eq}

Let's say , {eq}{T_A} {/eq} is the tension in the cable {eq}AC {/eq}, {eq}{T_B} {/eq} is the tension in the cable {eq}BC {/eq} and {eq}W {/eq} is the weight of bird.

Apply Lami's theorem to find the tension in the wire.

{eq}\dfrac{{{T_A}}}{{\sin \left( {90^\circ + \theta } \right)}} = \dfrac{{{T_B}}}{{\sin \left( {90^\circ + \theta } \right)}} = \dfrac{W}{{\sin 2\phi }}\\ \dfrac{{{T_A}}}{{\sin \left( {90^\circ + \theta } \right)}} = \dfrac{{{T_B}}}{{\sin \left( {90^\circ + \theta } \right)}} = \dfrac{{mg}}{{\sin 2\phi }} \\ T_A = T_B = \dfrac{{mg}}{{\sin 2\phi }} \\ T_A = T_B = \dfrac{{ 1 \times 9.81 }}{{\sin (2 \times 89.542^\circ) }} \\ \color {blue} {\boxed{\bf{T_A = T_B = 613.62 \ N}}} {/eq}

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Chapter 5 / Lesson 5Force is everywhere and it comes in a variety of sizes, directions, and types. In this video lesson, you'll identify force as well the different types of force that objects may experience.