Question

A bird has a mass of 26 g and perches in the middle of a stretched telephone line. (a) Show that the tension in the line can be calculated using the equation T = mg 2 sin θ . Determine the tension when (b) θ = 5° and (c) θ = 0.5° . Assume that each half of the line is straight.

Answer 1

Answer:

a) is demonstrated below, b) T=1.462N, c) T=14.6N

Step-by-step explanation:

a) Refer to the attached diagram.

Since the bird is standing in the middle of the line and each half is a straight line, Ta=Tb, so we will call the tension T=Ta=Tb and Tay=Tby

By trigonometry Tay=Ta·Sinθ

Since the system is in equilibrium W=Tay+Tby then:

W=2·Tay=2·Ta·Sinθ=2·T·Sinθ

Since W=mg, being m the mass of the bird and g, gravity:

$mg=2TSin\theta$

Isolating T, we demonstrate that

$T=\frac{mg}{2Sin\theta}$

b) Replacing θ=5º, m=0.026kg and g=9.8m/s² in the last equation, we can get the tension in Newtons:

$T=\frac{0.026*9.8}{2Sin5}=1.462N$

c) With θ=0.5º

$T=\frac{0.026*9.8}{2Sin0.5}=14.6N$