Question

A pendulum has an angular amplitude $\theta$ . The tension in the string at extreme position is T₁ and at the bottom is T₂. If it is known that T₂ = 2T₁, then what is the value of $cos\theta$ ?

Answer 1

Answer: 0.75

${\text{T}}_1=\text{mg cos}\theta$ ..... (1)
${\mathrm{T}}_2-\mathrm{mg}=\frac{{\mathrm{mv}}^2}{l}=\frac{\mathrm{m}}{l}(2\mathrm{gh})$
$=\frac{2\text{mg}}{\text{l}}\left(1-\text{cos}\theta \right)\text{l}$
$=2\text{mg}\left(1-\text{cos}\theta \right)\text{l}$
or ${\mathrm{T}}_2=\mathrm{mg}+2\mathrm{mg}(1-\cos \theta )$ ....... (2)

Given T₂ = 2T₁
or $\text{mg}+2\text{mg}\left(1-\text{cos}\theta \right)=\text{2mg cos}\theta$
or $\text{cos}\theta =\frac{3}{4}$
= 0.75