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.

Free-Body Diagram


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}


Learn more about this topic:

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Force: Definition and Types

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Chapter 5 / Lesson 5
78K

Force 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.


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