Question

The maximum tension in the string of an oscillating pendulum is double of the minimum tension. Find the angular amplitude.

Answer 1

The tension in the pendulum is maximum at the mean position and minimum on the extreme position.

Here $\frac{1}{2}m{v}^2-0=mgl(1-cos\theta )$

$v^2=2gl(1-cos\theta )...\left(1\right)$

Now, $T_{max}=mg+2mg(1-cos\theta )$

$[T=mg+(m{v}^2/l\left)\right]$

Again, $T_{min}=mgcos\theta$

According to question,

$T_{max}=2Tmin$

$\Rightarrow mg+2mg-2mgcos\theta =2mgcos\theta$

$\Rightarrow 3mg=4mgcos\theta$

$\Rightarrow cos\theta =\frac{3}{4}$

$\Rightarrow \theta =co{s}^{-1}\frac{3}{4}$