Question

The path length of one oscillation of a simple pendulum of length of $1$ meter is $16cm$ . Its maximum velocity is ( $g={\pi }^{2}m{s}^{-2}$ )
 $$\begin{gathered} \left(A\right)2\pi m{s}^{-1} \\ \left(B\right)4\pi m{s}^{-1} \\ \left(C\right)8\pi m{s}^{-1} \\ \left(D\right)16\pi m{s}^{-1} \end{gathered}$$

Answer 1

Hint :In order to solve this question, we are going to draw a schematic diagram of the pendulum and then calculate the height $h$ of the pendulum bob from the length $AOB$ and then, by equating the potential and the kinetic energies of the pendulum bob, the maximum velocity is calculated.
The formula used in this question is
The angle $\theta ={\displaystyle \frac{arc}{radius}}$
All of the potential energy at $A$ is converted to the kinetic energy at $O$
 $$\begin{gathered} P.E.=K.E. \\ mg\mathrm{\Delta }x={\displaystyle \frac{1}{2}}m{v}^2 \end{gathered}$$

Complete Step By Step Answer:
 Let us first draw the figure for the given arrangement

As it is given that the length of the complete oscillation is $16cm$ so, the length of $AOB$ will be $8cm$ , Now the angle $\theta ={\displaystyle \frac{arc}{radius}}$
Putting the values of the arc and the radius
 $\theta ={\displaystyle \frac{8cm}{100cm}}=0.08$
So, the angles ${\displaystyle \frac{\theta }{2}}$ equals $0.04$ radians
So, the height, $h$ is calculated as
 $$\begin{gathered} h=100cm\times \cos {\displaystyle \frac{\theta }{2}}=100cm\times \cos \left(0.04\right) \\ \Rightarrow h=100cm\times 0.9992=99.92cm \end{gathered}$$
Therefore, $\mathrm{\Delta }x=100cm-h=100-99.92=0.08cm$
Now, all of the potential energy at $A$ is converted to the kinetic energy at $O$
 $P.E.=K.E.$
So,
 $mg\mathrm{\Delta }x={\displaystyle \frac{1}{2}}m{v}^2$
Now, as it is given that, $g=x^2={\displaystyle \frac{1}{2}}{v}^2=100{\pi }^{2}cm{s}^{-2}$
Since, $1m=100cm$
Therefore,
 $$\begin{gathered} g\mathrm{\Delta }x={\displaystyle \frac{1}{2}}{v}^2 \\ {v}^2=2g\mathrm{\Delta }x \\ {v}^2=2\times 100{\pi }^{2}cm{s}^{-2}\times 0.08cm \\ {v}^2=16{\pi }^{2}c{m}^2{s}^{-2} \\ v=4\pi cm{s}^{-1} \end{gathered}$$

Note :
The angle $\theta$ is divided equally by the perpendicular bisector to the line segment. As the pendulum bob moves from a certain height to the reference level, all of its potential energy due to a particular height gets converted to the kinetic energy and when the bob goes from the reference level to the extreme point, the kinetic energy gets converted to the potential energy , this gives an energy equivalence.