Question

A bird has a mass of 26 g and perches in the middle of a stretched telephone line. Show that the tension in the line can be calculated using the equation T=mg/2sin theta

Answer 1

Let $\theta$ be an angle between the horizontal and the line. Let $\vec{T}_1, \vec{T}_2$ be the tensions directed from the bird along the line parts. The projections of these tensions on the vertical axis should compensate the weight of the bird. The sum of projections is $T_1\sin\theta + T_2\sin\theta = 2T\sin\theta.$

So $2T\sin\theta = mg$ and $T = \dfrac{mg}{2\sin\theta } = \dfrac{0.026\,\mathrm{kg}\cdot9.8\,\mathrm{N/kg}}{2\sin\theta} = \dfrac{0.127}{\sin\theta}.$

To calculate the precise value of tension we should know the value of $\theta.$