Question

A pendulum has an angular amplitude $\theta$. The tension in the string at extreme position is $T_1$ and at the bottom is $T_2$. If it is known that $T_2=2 T_1$, then what is the value of $100 \cos \theta$ ?

Answer 1

$T_1 = mg\cos\theta .....$(1)
$T _{2}- mg =\frac{ mv ^{2}}{l}=\frac{ m }{l}(2 gh ) $
$=\frac{2 mg }{l}(1-\cos \theta) l $
$=2 mg (1-\cos \theta) l$
or $T _{2}= mg +2 mg (1-\cos \theta)...(2)$

Given $T_2 = 2T_1$
or $\text{mg} + 2 \text{mg} \left(1 - \text{cos} \theta \right) = \text{2mg cos} \theta $
or $\text{cos} \theta = \frac{3}{4}$
$= 0.75$