A simple pendulum oscillates in a vertical plane. When it passes through the mean position, the tension in the string is 3 times the weight of the pendulum bob. What is the maximum displacement of the pendulum of the string with respect to the vertical ?

• A

  $30^{\circ}$
• B

  $45^{\circ}$
• C

  $60^{\circ}$
• Correct answer
  D

  $90^{\circ}$

Solution

In mean position,
$\begin{aligned} T & =3 m g \\ \text { Net force, } T-m g & =\frac{m v^2}{l}\end{aligned}$ $\begin{aligned} & 3 m g-m g \\ \Rightarrow \quad & \frac{m v^2}{l} \\ v & =\sqrt{2 g l} \end{aligned}$
Let maximum displacement with respect to vertical is $\theta$, then $\begin{aligned} \frac{1}{2} m v^2 & =m g h \\ & =m g(l-l \cos \theta) \\ \frac{1}{2} m(2 g l) & =m g l(1-\cos \theta) \\ \cos \theta & =0 \end{aligned}$ $\Rightarrow \quad \theta=90^{\circ}$

Hence, the correct answer is option: 'D'.